Quiz: Group and Vector Space Algebra — 59 questions

1. A matrix of format $(n,p)$ over $K$ has which dimensions?

A. $p$ rows and $n$ columns

B. $n$ diagonal entries and $p$ off-diagonal entries

C. $n+p$ rows and columns

D. $n$ rows and $p$ columns

Answer: (D)

Explanation

The format $(n,p)$ specifies $n$ rows followed by $p$ columns, with entries in $K$. Confusing the order would reverse the matrix dimensions rather than describe its stated format.

2. Which condition characterizes a square matrix of order $n$?

A. It has $n$ rows and $n$ columns

B. It has equal diagonal and off-diagonal entries

C. It has $n$ rows and any positive number of columns

D. It contains exactly $n$ nonzero entries

Answer: (A)

Explanation

A square matrix of order $n$ has the same number, $n$, of rows and columns. A rectangular matrix can have unequal dimensions, so merely having $n$ rows does not establish squareness.

3. What distinguishes the identity matrix $I_n$ from a general square matrix of order $n$?

A. Its diagonal entries are all $1$

B. Every entry in the matrix equals its row index

C. Its entries outside the diagonal are all $1$

D. Its diagonal entries are all equal to the matrix order

Answer: (A)

Explanation

The identity matrix is diagonal and has a $1$ in every diagonal position. Its off-diagonal entries are zero, so assigning them the value $1$ describes a different matrix.

4. For matrices $A\in M_{np}(K)$ and $B\in M_{pq}(K)$, when is the product $AB$ defined?

A. When $A$ and $B$ have the same number of rows

B. When the columns of $A$ equal the rows of $B$

C. When $A$ and $B$ are both square matrices

D. When the rows of $A$ equal the columns of $B$

Answer: (B)

Explanation

The inner dimensions must match: $A$ has $p$ columns and $B$ has $p$ rows. Matching the opposite dimensions concerns the possible product $BA$ and does not guarantee that $AB$ exists.

5. If $A$ has format $(n,p)$, what is the format of its transpose $^{t}A$?

A. $(n,p)$

B. $(n,n)$

C. $(p,n)$

D. $(p,p)$

Answer: (C)

Explanation

Transposition turns the columns of $A$ into rows, changing $(n,p)$ into $(p,n)$. The original format remains $(n,p)$, so retaining it would not describe the transpose in general.

6. What is the trace of a square matrix $A$ of order $n$?

A. The sum of its diagonal entries, $\operatorname{Tr}(A)=\sum_{i=1}^{n}a_{ii}$

B. The sum of entries in its first row, $\sum_{j=1}^{n}a_{1j}$

C. The sum of all entries in the matrix, $\sum_{i,j}a_{ij}$

D. The product of its diagonal entries, $\prod_{i=1}^{n}a_{ii}$

Answer: (A)

Explanation

The trace adds the entries whose row and column indices are equal, namely the diagonal terms. Summing every entry or multiplying diagonal entries defines different operations.

7. For a $2\times2$ matrix \begin{pmatrix}a&b\c&d\end{pmatrix}, which expression gives its determinant?

A. $ad+bc$

B. $ad-bc$

C. $ab-cd$

D. $ac-bd$

Answer: (B)

Explanation

The determinant of a $2\times2$ matrix is formed by multiplying the main diagonal entries and subtracting the product of the other diagonal. Replacing subtraction with addition gives a different expression, not the determinant.

8. When expanding a $3\times3$ determinant along a row or column, what is multiplied by each signed entry?

A. Its corresponding minor

B. The matrix trace

C. The entire deleted column

D. The entire deleted row

Answer: (A)

Explanation

A determinant expansion combines each entry with its corresponding minor and applies the appropriate sign. A minor is obtained by deleting a row and column, whereas a cofactor includes the sign factor as well.

9. What is the determinant of a triangular matrix with diagonal entries $d_1,\ldots,d_n$?

A. $n\prod_{i=1}^{n}d_i$

B. $d_1-d_n$

C. $\sum_{i=1}^{n}d_i$

D. $\prod_{i=1}^{n}d_i$

Answer: (D)

Explanation

For a triangular matrix, the determinant equals the product of its diagonal entries. The sum of those entries is the trace, which is a different matrix quantity.

10. How does adding a linear combination of other rows to one row affect a determinant?

A. It multiplies the determinant by $-1$

B. It multiplies the determinant by the combination's coefficient

C. It leaves the determinant unchanged

D. It makes the determinant equal to zero

Answer: (C)

Explanation

Adding a linear combination of other rows to a row preserves the determinant. Multiplication by $-1$ occurs when two rows or two columns are exchanged, not when such a row operation is performed.

11. Which condition defines a square matrix A as invertible?

A. There is a matrix B such that AB = In, without requiring BA = In

B. The matrix A has equal row sums and column sums

C. There is a square matrix B such that AB = In = BA

D. The matrix A has at least one nonzero entry in every row

Answer: (C)

Explanation

A is invertible when it has a two-sided inverse B satisfying both AB = In and BA = In. Requiring only one product to equal the identity does not state the defining two-sided inverse condition.

12. A square matrix has determinant equal to zero. What conclusion follows about the matrix?

A. It has a two-sided inverse after row rearrangement

B. It is not invertible

C. It has an inverse with determinant zero

D. It is invertible because its determinant is defined

Answer: (B)

Explanation

A square matrix is invertible if and only if its determinant is nonzero, so determinant zero means the matrix is not invertible. A determinant-zero matrix cannot have a two-sided inverse.

13. For an invertible square matrix A, which formula gives its inverse?

A. $A^{-1}=\det(A)\,{}^{t}\!\operatorname{com}(A)$

B. $A^{-1}=\frac{1}{\det A}\,{}^{t}\!\operatorname{com}(A)$

C. $A^{-1}=\frac{1}{\det A}\,\operatorname{com}(A)$

D. $A^{-1}=\det(A)\,{}^{t}\!A$

Answer: (B)

Explanation

The inverse is obtained by multiplying the transpose of the cofactor matrix by the reciprocal of the determinant. Omitting the transpose or replacing the reciprocal with the determinant changes the formula.

14. In the matrix equation $AX=B$ for a linear system, what does each symbol represent?

A. A is the right-hand-side vector, X is the unknown vector, and B is the coefficient matrix

B. A is the unknown vector, X is the coefficient matrix, and B is the right-hand-side vector

C. A is the coefficient matrix, X is the right-hand-side vector, and B is the unknown vector

D. A is the coefficient matrix, X is the unknown vector, and B is the right-hand-side vector

Answer: (D)

Explanation

The matrix form separates the coefficients, unknowns, and constants as A, X, and B respectively. Confusing X with B reverses the roles of the unknown vector and the right-hand-side vector.

15. A linear system has rank r and p unknowns with r < p. How many unknowns are treated as parameters?

A. $r-p$

B. $p-r$

C. $p+r$

D. $r$

Answer: (B)

Explanation

When the rank is smaller than the number of unknowns, the difference $p-r$ gives the number of parameters. The rank itself counts independent coefficient information rather than free unknowns in this case.

16. What is the rank of a matrix?

A. The number of columns in the matrix after row reduction

B. The largest order of a square submatrix with nonzero determinant

C. The sum of all entries in a largest square submatrix

D. The number of equations in the associated linear system

Answer: (B)

Explanation

The rank is the maximum size of a square submatrix whose determinant is nonzero. The number of columns or equations can differ from the rank and does not define it.

17. In Cramer's rule for $AX=B$ with $\det(A)\ne0$, how is $x_i$ computed?

A. $x_i=\frac{D_{x_i}}{\det A}$, where column i of A is replaced by B

B. $x_i=\frac{\det A}{D_{x_i}}$, where row i of A is replaced by B

C. $x_i=\frac{D_{x_i}}{\det A}$, where column i of A is replaced by X

D. $x_i=D_{x_i}\det A$, where column i of B is replaced by A

Answer: (A)

Explanation

Cramer's rule divides the determinant formed by replacing column i of A with B by det(A). Replacing a row or using the unknown vector X does not produce the required determinant.

18. What makes a binary operation an internal composition law on a nonempty set E?

A. It maps pairs from E to elements outside E when needed

B. It maps every pair in $E\times E$ to an element of E

C. It combines elements of E with scalars from another set

D. It maps every element of E to a pair in $E\times E$

Answer: (B)

Explanation

An internal composition law is a function from $E\times E$ into E, so combining two elements of E produces another element of E. An operation that leaves E is external rather than internal.

19. A subset A of E is stable under an internal law ⋆ when which condition holds?

A. For every x,y in A, $x\star y\in E\setminus A$

B. For every x in A, there is a y in E with $x\star y\notin A$

C. For every x,y in A, $x\star y\in A$

D. For every x,y in E, $x\star y\in A$

Answer: (C)

Explanation

Stability means that applying the law to any two members of A remains within A. Requiring products of all elements of E to lie in A is a stronger and different condition.

20. Which equation expresses associativity of a law ⋆ on E?

A. $a\star b=b\star a$ for every a,b in E

B. $a\star e=a=e\star a$ for every a in E

C. $a\star(b\star c)=c\star(b\star a)$ for every a,b,c in E

D. $a\star(b\star c)=(a\star b)\star c$ for every a,b,c in E

Answer: (D)

Explanation

Associativity states that changing the grouping of three factors does not change the result. The equation $a\star b=b\star a$ expresses commutativity, not associativity.

21. What property must an identity element e satisfy for a law ⋆ on E?

A. $a\star e=e=e\star a$ for every a in E

B. $a\star e=a=e\star a$ for every a in E

C. $a\star a=e$ for every a in E

D. $e\star e=a$ for every a in E

Answer: (B)

Explanation

An identity element leaves every element unchanged when placed on either side of the operation. The condition that products equal e describes an inverse-like relation rather than an identity element.

22. Which additional property distinguishes an abelian group from a general group?

A. Its elements lack individual inverses

B. Its operation is defined externally

C. Its operation is commutative

D. Its operation has no identity element

Answer: (C)

Explanation

An abelian group is a group whose operation is commutative, so the order of two factors does not affect their product. A general group still has an identity and inverses, so lacking those properties would violate the definition of a group.

23. Which condition is required for a nonempty subset H of a group G to be a subgroup?

A. H is an arbitrary collection closed under external maps

B. H contains a different operation from the one on G

C. H contains at least two elements of G

D. H contains the identity and inverses of its elements

Answer: (D)

Explanation

A subgroup must form a group under the operation induced from G, which requires the identity, closure, and inverses. An arbitrary subset can fail to contain the identity or fail to be closed under the induced operation.

24. For a group morphism f from G to F, what does the kernel consist of?

A. Elements of G mapped to every element of F

B. Elements of F that have no preimage in G

C. All values attained by f in the group F

D. Elements of G mapped to the identity of F

Answer: (D)

Explanation

The kernel is the set of elements in the domain G whose images equal the identity element of F. The set of all values attained by f is instead its image, while elements without preimages describe a different property.

25. What set is the image of a group morphism f:G→F?

A. The elements of G that remain fixed under f

B. The values f(x) attained as x ranges over G

C. The elements of F having exactly one preimage

D. The elements of G mapped to the identity of F

Answer: (B)

Explanation

The image is the subset of F consisting of all values attained by applying f to elements of G. The elements mapped to the identity form the kernel, not the image.

26. Why is a nonempty subset H of a group G not automatically a subgroup?

A. It may use the same operation as G on its elements

B. It may contain too many elements to inherit a group structure

C. It may lack the identity or fail to be closed under the induced law

D. It may contain elements that also belong to other subsets of G

Answer: (C)

Explanation

A subgroup must be a group under the operation inherited from G, including the identity, closure, and inverses. Nonemptiness alone does not guarantee these properties, even when the subset uses the same operation.

27. Which collection of conditions characterizes a subgroup H of a group G?

A. H is closed, contains the identity, and contains every required inverse

B. H contains inverses but need not be closed under the group operation

C. H is nonempty and contains products but no identity requirement

D. H contains the identity and products but may omit inverses

Answer: (A)

Explanation

The subgroup criterion requires closure under the group operation, the identity element, and the inverse of every element in H. Closure alone or combining closure with just one of the other properties is insufficient.

28. Which one-step test can determine whether a nonempty subset H of G is a subgroup?

A. For every x,y in H, verify that $x\star y$ belongs to G

B. For every x,y in H, verify that $x\star y^{-1}$ belongs to H

C. For every x in H, verify that $x^{-1}$ belongs to G

D. For every x,y in G, verify that $x\star y^{-1}$ belongs to H

Answer: (B)

Explanation

A nonempty subset H is a subgroup exactly when it contains $x\star y^{-1}$ for every pair x,y in H. The other tests either check membership in the larger group or quantify over elements outside H, so they do not establish the subgroup criterion.

29. What defining equation must a group morphism f satisfy for elements x and y of G?

A. $f(x\mathbin{\bullet}y)=f(x)+f(y)$

B. $f(x\mathbin{\bullet}y)=f(x)\mathbin{\star}f(y)$

C. $f(x\mathbin{\bullet}y)=f(x)f(y)$

D. $f(x\mathbin{\bullet}y)=f(x)\mathbin{\bullet}f(y)$

Answer: (B)

Explanation

A group morphism preserves the specified group operations, so the image of the product under the operation on G equals the product of the images under the operation on F. The other expressions impose operations that may not be the laws of the two groups.

30. If f:G→F is a group morphism, what does it do to the identity and inverses?

A. It maps the identity to an arbitrary element and reverses products

B. It maps every element to the identity and preserves products

C. It maps the identity to the identity and inverses to inverses

D. It preserves each element itself and may alter its inverse

Answer: (C)

Explanation

Operation preservation forces $f(e_G)=e_F$ and $f(x^{-1})=f(x)^{-1}$. A morphism need not preserve elements themselves, because its values can identify distinct elements.

31. For a group morphism f:G→F, what condition is equivalent to injectivity?

A. Its image is $F$

B. Its image contains the identity of F

C. Its kernel is all of G

D. Its kernel is $\{e_G\}$

Answer: (D)

Explanation

A group morphism is injective exactly when the only element mapped to the identity of F is $e_G$. Having image equal to F characterizes surjectivity, not injectivity.

32. When is a group morphism f:G→F an isomorphism?

A. When its image is a subgroup without requiring injectivity

B. When it preserves the identity but may fail to preserve products

C. When its kernel is G and its image is F

D. When its kernel is $\{e_G\}$ and its image is F

Answer: (D)

Explanation

An isomorphism is a morphism that is both injective and surjective, equivalent here to having trivial kernel and image equal to F. Preserving the identity or having a subgroup as the image does not ensure bijectivity.

33. Which condition is required in the definition of a $K$-vector space?

A. Scalar multiplication must map vectors to elements of the scalar field

B. The addition operation must make the set a group with finitely many elements

C. The addition operation must make the set an abelian group

D. Every pair of vectors must have a unique scalar multiple

Answer: (C)

Explanation

A vector space requires an abelian group under addition together with scalar multiplication satisfying the vector-space axioms. Being a group does not provide the required scalar multiplication structure, and finiteness is not required.

34. Over which scalar fields can $\mathbb{C}^n$ be regarded as a vector space?

A. Over a scalar field determined by the value of $n$

B. Over both $\mathbb{C}$ and $\mathbb{R}$

C. Over $\mathbb{C}$ but not over $\mathbb{R}$

D. Over $\mathbb{R}$ but not over $\mathbb{C}$

Answer: (B)

Explanation

The complex coordinate space $\mathbb{C}^n$ supports scalar multiplication by complex numbers and therefore also by real numbers viewed as complex scalars. The choice of scalar field is not determined by the dimension.

35. Which expression is a linear combination of vectors $u_1,\ldots,u_p$?

A. $\lambda_1u_1+\cdots+\lambda_pu_p$ with scalars $\lambda_i$

B. A vector selected from the list $u_1,\ldots,u_p$ without coefficients

C. A sum of listed vectors whose coefficients must all equal one

D. A product of two listed vectors using the vector multiplication operation

Answer: (A)

Explanation

A linear combination assigns scalar coefficients to the listed vectors and adds the resulting scalar multiples. A listed vector or an unweighted sum is not the general definition of a linear combination.

36. What does it mean for a family of vectors to span a vector space $E$?

A. The family contains no vector that can be expressed using the others

B. The family has the same number of vectors as the dimension of $E$

C. Every vector in the family has a unique coordinate representation

D. Every vector in $E$ can be expressed as a linear combination of the family

Answer: (D)

Explanation

A spanning family reaches every vector of $E$ through linear combinations of its members. Linear independence concerns redundancy, while equal cardinality and uniqueness require additional conditions.

37. When is a family $(u_1,\ldots,u_q)$ linearly independent?

A. When the equation $\lambda_1u_1+\cdots+\lambda_qu_q=0_E$ forces every coefficient to be zero

B. When every vector in the ambient space is a linear combination of the family

C. When the family contains more vectors than the dimension of the ambient space

D. When each vector has a nonzero coefficient in at least one combination

Answer: (A)

Explanation

Linear independence means that the zero vector has no nontrivial representation using the family. Spanning the ambient space is a separate property, and having too many vectors generally creates dependence in finite dimensions.

38. If $B=(e_1,\ldots,e_n)$ is a basis of $E$, what is true about each vector $u\in E$?

A. It can be represented by a linear combination, but the coefficients may vary

B. It can be represented only when all coefficients are nonzero

C. It must equal one of the basis vectors

D. It has a unique representation $u=\alpha_1e_1+\cdots+\alpha_ne_n$

Answer: (D)

Explanation

A basis is both spanning and linearly independent, so every vector has a representation and independence makes its coefficients unique. A vector need not itself be a basis vector, and some coordinates may be zero.

39. Which property must a nonempty subset $F$ have to be a vector subspace of $E$?

A. It must be closed under both vector addition and scalar multiplication

B. It must contain every scalar multiple of vectors outside the subset

C. It must be closed under vector addition while scalar multiplication may leave it

D. It must contain a basis of the whole space and be finite

Answer: (A)

Explanation

A subspace inherits both operations from the ambient vector space, so it must remain closed under addition and scalar multiplication. Closure under just one operation does not establish a vector subspace.

40. Which criterion is sufficient for a nonempty subset $F\subseteq E$ to be a vector subspace?

A. For all $u,v\in F$, the ordinary sum $u+v$ belongs to $F$

B. For all $u,v\in F$ and $\alpha,\beta\in K$, $\alpha u+\beta v\in F$

C. For every $u\in F$, at least one nonzero scalar multiple of $u$ belongs to $F$

D. For every scalar $\alpha\in K$, some vector of $F$ has scalar multiple $\alpha u$ in $F$

Answer: (B)

Explanation

The combined condition includes closure under addition and under arbitrary scalar multiplication, and together with nonemptiness it characterizes a subspace. Checking ordinary addition alone does not guarantee scalar closure.

41. What set is represented by the notation $\operatorname{vect}(A)$ for $A=\{u_1,\ldots,u_p\}$?

A. The set containing precisely the vectors listed in $A$

B. The set of all linear combinations of $u_1,\ldots,u_p$

C. The set of vectors linearly independent from every member of $A$

D. The set of vectors having the same norm as at least one member of $A$

Answer: (B)

Explanation

The subspace generated by $A$ consists of every linear combination of its vectors. It generally contains more vectors than the original set and is defined through scalar combinations, not norms or independence from the set.

42. If $\dim E_1=5$, $\dim E_2=4$, and $\dim(E_1\cap E_2)=2$, what is $\dim(E_1+E_2)$?

A. $3$

B. $7$

C. $9$

D. $6$

Answer: (B)

Explanation

The dimension formula gives $\dim(E_1+E_2)=\dim E_1+\dim E_2-\dim(E_1\cap E_2)=5+4-2=7$. The intersection dimension must be subtracted because shared directions are counted in both subspaces.

43. Which condition makes a map $f:E\to F$ linear?

A. It sends every basis of the domain to a basis of the codomain

B. It maps every vector to a nonzero vector in the codomain

C. It preserves vector addition and scalar multiplication

D. It preserves vector lengths and angles between vectors

Answer: (C)

Explanation

A linear map satisfies $f(u+v)=f(u)+f(v)$ and $f(\lambda u)=\lambda f(u)$ for all relevant vectors and scalars. Preserving lengths or sending bases to bases is not required for linearity.

44. How are the columns of the matrix of $f:E\to F$ determined when the domain basis is $B=(e_1,\ldots,e_p)$ and the codomain basis is $U=(u_1,\ldots,u_n)$?

A. They are the coordinate vectors of $f(e_1),\ldots,f(e_p)$ in basis $U$

B. They are the coordinate vectors of the basis vectors of $E$ in basis $U$

C. They are the coordinate vectors of $f(u_1),\ldots,f(u_n)$ in basis $B$

D. They are the coordinates of the kernel vectors expressed in basis $B$

Answer: (A)

Explanation

Each column records the coordinates, in the codomain basis $U$, of the image of one domain basis vector. The other choices confuse domain vectors, codomain basis vectors, or kernel information with the matrix construction.

45. If $f:E\to F$ and $g:F\to G$ are composable linear maps, which matrix product represents $g\circ f$?

A. $\operatorname{mat}_{B,V}(g)\operatorname{mat}_{U,B}(f)$

B. $\operatorname{mat}_{U,B}(f)\operatorname{mat}_{V,U}(g)$

C. $\operatorname{mat}_{B,U}(f)\operatorname{mat}_{U,V}(g)$

D. $\operatorname{mat}_{U,V}(g)\operatorname{mat}_{B,U}(f)$

Answer: (D)

Explanation

The matrix of the composition is obtained by applying the matrix of $f$ first and then the matrix of $g$, giving $\operatorname{mat}_{U,V}(g)\operatorname{mat}_{B,U}(f)$. Reversing the order generally produces an undefined or incorrect product.

46. A linear map $f:E\to F$ has a three-dimensional domain and a two-dimensional image; what is $\dim(\ker f)$?

A. $5$

B. $3$

C. $1$

D. $2$

Answer: (C)

Explanation

The rank theorem gives $\dim E=\dim(\ker f)+\dim(\operatorname{Im}f)$, so $3=\dim(\ker f)+2$ and the kernel has dimension $1$. The image dimension is the rank, not the kernel dimension.

47. When is a scalar $\lambda$ an eigenvalue of an endomorphism $f$?

A. When the zero vector satisfies $f(u)=\lambda u$

B. When every vector satisfies $f(u)=\lambda u$

C. When some nonzero vector satisfies $f(u)=\lambda u$

D. When some nonzero vector satisfies $f(u)=u+\lambda$

Answer: (C)

Explanation

An eigenvalue is associated with a nonzero eigenvector satisfying $f(u)=\lambda u$. The zero vector cannot identify an eigenvalue because it satisfies that equation for every scalar.

48. Which set is the eigenspace associated with an eigenvalue $\lambda$ of $f:E\to E$?

A. $\operatorname{Im}(f-\lambda\operatorname{Id}_E)$

B. $\ker(f+\lambda\operatorname{Id}_E)$

C. $\operatorname{Im}(f)+\lambda E$

D. $\ker(f-\lambda\operatorname{Id}_E)$

Answer: (D)

Explanation

The eigenspace is $E_\lambda=\ker(f-\lambda\operatorname{Id}_E)$, consisting of all vectors satisfying $f(u)=\lambda u$. It includes the zero vector, unlike the set of individual eigenvectors.

49. For a square matrix $A$ of order $n$, which expression defines its characteristic polynomial?

A. $P_A(\lambda)=\det(A) - \lambda I_n$

B. $P_A(\lambda)=\det(A+\lambda I_n)$

C. $P_A(\lambda)=\operatorname{tr}(A-\lambda I_n)$

D. $P_A(\lambda)=\det(A-\lambda I_n)$

Answer: (D)

Explanation

The characteristic polynomial is defined as the determinant $\det(A-\lambda I_n)$. A trace expression and an uncombined determinant-minus-matrix expression do not give the characteristic polynomial.

50. An endomorphism on an $n$-dimensional space has $n$ distinct eigenvalues; what conclusion follows?

A. It is diagonalizable because eigenvectors for those eigenvalues form a basis

B. It has a repeated eigenvalue because its space has dimension $n$

C. It is nilpotent because its eigenvalues are distinct

D. It is singular because each eigenvalue has a separate eigenspace

Answer: (A)

Explanation

Distinct eigenvalues yield linearly independent eigenvectors, and $n$ such vectors form a basis of an $n$-dimensional space, making the endomorphism diagonalizable. Nilpotence and singularity do not follow from distinctness of the eigenvalues.

51. What does it mean for a polynomial $P\in K[X]$ to annihilate a matrix $A\in M_n(K)$?

A. It is defined by the equation $P(X)=\det(A-XI_n)$

B. It satisfies $\det(P(A))=P(\det A)$

C. It satisfies $P(A)=I_n$

D. It satisfies $P(A)=0_n$

Answer: (D)

Explanation

An annihilating polynomial is one whose evaluation at the matrix gives the zero matrix, $P(A)=0_n$. The determinant formula defines a characteristic polynomial, which is a different concept.

52. For $A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, why is $P(X)=X^3-X^2-X+I_2$ annihilating?

A. Because $A^2=A$ and $A^3=I_2$

B. Because $A^2=0_2$ and $A^3=0_2$

C. Because $A^2=-I_2$ and $A^3=-A$

D. Because $A^2=I_2$ and $A^3=A$

Answer: (D)

Explanation

Substituting the matrix gives $P(A)=A^3-A^2-A+I_2=A-I_2-A+I_2=0_2$, using $A^2=I_2$ and $A^3=A$. Those identities depend on this particular matrix and do not hold for arbitrary matrices.

53. Which statement correctly characterizes a diagonalizable matrix?

A. It is already diagonal in the basis used to write its entries.

B. It is similar to a diagonal matrix, possibly after changing the basis.

C. It has distinct entries along its main diagonal in its original form.

D. It has a triangular form in every basis of the vector space.

Answer: (B)

Explanation

A matrix is diagonalizable when a change of basis makes it diagonal, which means it is similar to a diagonal matrix. Being diagonal in the original basis is stronger and is not required.

54. What condition is equivalent to an endomorphism being diagonalizable?

A. The characteristic polynomial has one repeated root for each eigenspace.

B. The vector space contains at least one eigenvector for each eigenvalue.

C. The matrix has a triangular representation in some selected basis.

D. The vector space has a basis consisting entirely of eigenvectors.

Answer: (D)

Explanation

Diagonalizability is equivalent to the existence of a basis made entirely of eigenvectors. Having some eigenvectors, even one for each eigenvalue, may not provide enough independent vectors for a basis.

55. An operator on an n-dimensional vector space has n distinct eigenvalues. What conclusion follows?

A. The operator is diagonalizable because the corresponding eigenvectors are linearly independent.

B. The operator cannot be diagonalized because its eigenvalues have different values.

C. The operator has repeated eigenspaces because each eigenvalue occurs once.

D. The operator is triangularizable but lacks enough eigenvectors for a diagonal form.

Answer: (A)

Explanation

Eigenvectors associated with distinct eigenvalues are linearly independent, so n distinct eigenvalues provide an eigenbasis in an n-dimensional space. Repeated eigenvalues do not give the same guarantee.

56. Which sequence correctly describes a method for diagonalizing an endomorphism?

A. Compute the determinant in one basis, list the matrix columns, and use them as eigenvectors for the new basis.

B. Find a triangular form first, discard dependent eigenvectors, and then reconstruct the characteristic polynomial.

C. Choose any basis, compute the trace, select a maximal independent family, and write a triangular representation.

D. Solve the characteristic equation, find eigenspace bases, form an eigenbasis, and write the diagonal representation.

Answer: (D)

Explanation

Diagonalization proceeds by finding eigenvalues, determining bases of their eigenspaces, assembling an eigenbasis, and expressing the endomorphism in that basis. A triangular representation does not by itself supply the required eigenbasis.

57. Which statement correctly characterizes a triangularizable matrix?

A. It is similar to a triangular matrix after a suitable change of basis.

B. It is triangular in every basis used to represent the associated endomorphism.

C. It is similar to a diagonal matrix with one eigenvalue in each position.

D. It has linearly independent eigenvectors spanning the whole vector space.

Answer: (A)

Explanation

Triangularizability means that some basis produces a triangular matrix, equivalently that the original matrix is similar to a triangular one. A diagonal form and a complete eigenbasis describe the stronger property of diagonalizability.

58. Over a field K, what condition is necessary and sufficient for an n-dimensional matrix to be triangularizable?

A. Its eigenspaces all have dimensions equal to their algebraic multiplicities over K.

B. Its matrix entries are arranged in triangular order in the original basis.

C. Its characteristic polynomial splits into n roots counted with multiplicity over K.

D. It has n distinct eigenvalues, each producing an independent eigenvector over K.

Answer: (C)

Explanation

A matrix over K is triangularizable exactly when its characteristic polynomial splits over K, with n roots counted according to multiplicity. Matching eigenspace dimensions is an additional requirement for diagonalizability.

59. Which procedure correctly triangularizes an endomorphism?

A. Compute the characteristic polynomial, select arbitrary independent vectors, and use them as columns of a triangular matrix.

B. Solve the characteristic equation, choose independent eigenvectors, complete them to a basis, and write the triangular matrix.

C. Determine one eigenvector, repeat it across the basis, and record its eigenvalue in each diagonal position.

D. Find all eigenvectors, require them to form a full basis, and place their eigenvalues on a diagonal matrix.

Answer: (B)

Explanation

Triangularization uses eigenvectors to form a maximal independent family, extends that family to a full basis, and then expresses the map in the resulting basis. Requiring a complete eigenbasis would impose the stronger condition of diagonalizability.