Chapter 2: Matrix Algebra

Question 1

Matrix

Answer 1

a rectangular array of numbers called
the entries or elements of the matrix.

Question 2

For a matrix A we write the entry in row i, column j as …

Answer 2

a_ij

Question 3

if A is an m × n matrix then A has …

Answer 3

m rows and n columns

Question 4

Square matrix

Answer 4

if m=n

Question 5

diagonal matrix

Answer 5

iff its off-diagonal elements are zero (for i 6= j, aij = 0)

Question 6

The n × n identity matrix In is the diagonal
matrix with all diagonal entries equal to …

Answer 6

1

Question 7

Two matrices are equal if and only if

Answer 7

they have the same number of rows and the same number of columns and the corresponding entries are equal.

Question 8

If A = (aij) and B = (bij) are m × n matrices, then we define the new m × n matrix A + B componentwise:

Answer 8

A + B = (aij) + (bij) = (aij + bij);

Question 9

If A = (aij) is an m × n matrix and c is a scalar then we define the new m × n matrix cA componentwise:

Answer 9

cA = c(aij) = (caij).

Question 10

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(a) A + B =

Answer 10

B + A

Question 11

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(b) (A + B) + C =

Answer 11

A + (B + C).

Question 12

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(c) There is an m × n matrix 0 such that

Answer 12

A + 0 = A, for each A

Question 13

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(d) For each A there is an m × n matrix, −A such that

Answer 13

A + (−A) = 0.

Question 14

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(e) c(A + B) =

Answer 14

cA + cB

Question 15

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(f) (c + d)A =

Answer 15

cA + dA

Question 16

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(g) c(dA) =

Answer 16

(cd)A.

Question 17

Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(h) 1A =

Answer 17

A

Question 18

Transpose of an m × n matrix A. written A^T

Answer 18

the n × m matrix whose rows are the columns of A in the same order.

Question 19

If A = [aij] then A^T =

Answer 19

[aji].

Question 20

Theorem 2.1.2 Let A and B be matrices of the same size, and let k ∈ R
(a) (A^T)^T =

Answer 20

A

Question 21

Theorem 2.1.2 Let A and B be matrices of the same size, and let k ∈ R
(b) (A + B)^T =

Answer 21

A^T + B^T

Question 22

Theorem 2.1.2 Let A and B be matrices of the same size, and let k ∈ R
(c) (kA)^T =

Answer 22

k(A^T)

Question 23

Symmetric square matrix

Answer 23

if A^T = A.

(That is, A is equal to its own transpose)

Question 24

If A and B are square symmetric matrices, then A + B is

Answer 24

symmetric

Question 25

If A is a square matrix (not necessarily symmetric), then A + A^T is

Answer 25

a symmetric matrix.

Question 26

Ordered n-tuple

Answer 26

An ordered sequence (a1, a2, . . . , an) of real numbers

Question 27

Definition. Let R denote the set of all real numbers. The set of all ordered n-tuples from R is denoted by

Answer 27

R^n.

R^n = {(x1, x2, · · · , xn) : x1, · · · , xn ∈ R} .

Question 28

If x, y ∈ R^n it is clear that their matrix sum x + y is also in R^n as is the scalar multiple kx for any real number k. We express this observation by saying that

Answer 28

R^n is closed under addition and scalar
multiplication.

Question 29

Definition A vector v is a linear combination of vectors v1, · · · , vk if and only if

Answer 29

there are scalars c1, c2, · · · , ck such that
v = c1v1 + c2v2 + · · · + ckvk.

Question 29

Theorem 2.2.1
a) Every system of linear equations has the form

Answer 29

Ax = b where A is the coefficient matrix, b is the constant matrix, and x is the matrix of variables.

Question 30

Coefficients of the linear combination

Answer 30

The scalars c1, c2, · · · , ck in v = c1v1 + c2v2 + · · · + ckvk.

Question 31

Theorem 2.2.1
b) The system Ax = b is consistent if and only if

Answer 31

b is a linear combination of the columns of A

Question 32

Theorem 2.2.1
c) If a1, a2 . . . , an are the columns of A and if x = [x1,…xm] (as a column vector), then
x is a solution to the linear system Ax = b if and only if

Answer 32

x1, x2, . . . , xn are a solution of the vector equation x1a1 + x2a2 + · · · + xnan = b

Question 33

Definition. A function or mapping or transformation T from R^m to R^n is a

Answer 33

rule that assigns to each vector x ∈ R^m a unique vector T(x) ∈ R^n

Question 34

The domain of T in the transformation T from R^m to R^n is

Answer 34

R^m

Question 35

The codomain of T in the transformation T from R^m to R^n is

Answer 35

R^n

Question 36

To indicate that T is a map with domain R^m and codomain R^n, write

Answer 36

T: R^m –> R^n

Question 37

If x is in the domain of T, the image of x is

Answer 37

the vector T(x)

Question 38

The range of T in the transformation T from R^m to R^n is

Answer 38

the set of all possible images,

i.e. range(T) = {T(x) : x ∈ R^m}.

Question 39

Definition. If A is any m × n matrix, multiplication by A gives

Answer 39

a matrix transformation T_A : R^n → R^m defined by T_A(x) = Ax for every x in Rn

Question 40

Relection in x-axis is achived by multiplying by the matrix

Answer 40

l1 0 l (Check this with vector [x,y]
L0 -1J T([x,y])=[x,-y]

Question 41

Counterclockwise rotation about the origin through pi/2 radians

Answer 41

l0 -1l T([x,y])=[-y,x]
L1 0J

Question 42

x-expansion (compression)

Answer 42

la 0l T([x,y])=[ax,y]
L0 1J

For a > 1 we have expansion and for a < 1 we have compression

Question 43

x-shear

Answer 43

l1 al T([x,y])=[x+ay,y]
L0 1J

For a > 0 we have a positive x-shear and for a < 0 we have a negative x-shear

Question 44

The zero transformation

Answer 44

If A is the m × n zero matrix, T_0 : R^n → R^m is given by T_0(v) = 0R^m for all v ∈ R^n

Question 45

The identity transformation

Answer 45

If A is the n × n identity matrix, I_n : R^n → R^n given by I_n(v) = v for all v ∈ Rn

Question 46

Let T : R^k → R^m and S : R^n → R^m be transformations. The composition S ◦ T is the mapping S ◦ T : R^k → R^m defined by
(S ◦ T)(x) =

Answer 46

S(T(x))

for all x ∈ R^n.

Question 47

Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(f) (AB)^T =

Answer 47

B^TA^T

Question 48

Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(e) k(AB) =

Answer 48

(kA)B = A(kB)

Question 49

Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(d) (A + B)C =

Answer 49

AC + BC

Question 50

Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(c) A(B + C) =

Answer 50

AB + AC

Question 51

Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(b) A(BC) =

Answer 51

(AB)C

Question 52

Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(a) I_m A =

Answer 52

A = AI_n if A is m × n

Question 53

Inverse of A (n × n matrix)

Answer 53

an n × n matrix A’ such that AA’ = I_n, and A’A = I_n.

Question 54

A is invertible

Answer 54

if A’ exists

Question 55

A is singular or not invertible

Answer 55

If no inverse exists

Question 56

Theorem Uniqueness of the matrix inverse. If an n × n matrix A is invertible then

Answer 56

its inverse is unique.

Question 57

Theorem 2.4.2 If A is an invertible n × n matrix then the system of linear equations given by Ax = b has the unique solution given by

Answer 57

x = A^−1b.

Question 58

If A = [a b]
[c d]

then A is invertible if ______
in which case A^-1= _______

Answer 58

1. ad-bc =/= 0
2. 1/(ad-bc)*[d -b]
   [-c a]

Question 59

Gauss–Jordan method for computing the inverse.
If A is an invertible matrix, there exists a sequence of elementary row operations that carry A to the identity matrix I of the same size, written A → I. This same series of row operations carries I to A^−1. The algorithm can be summarised as follows:

Answer 59

[A | I] –ERO’s–> [I | A^-1]

Question 60

Theorem 2.4.4
(a) If A is an invertible matrix, then A^−1 is _____ and (A^−1)^−1 = ____

Answer 60

1. invertible
2. A.

Question 61

Theorem 2.4.4
(b) If A is an invertible matrix and c =/= 0 is a scalar then cA is ____ and (cA)^−1 = ______

Answer 61

1. invertible
2. 1/c*A^−1

Question 62

Theorem 2.4.4
(c) If A and B are invertible matrices of the same size, then AB is _____ and (AB)^−1 = _____

Answer 62

1. invertible
2. B^−1A^−1

Question 63

Theorem 2.4.4
(d) If A is an invertible matrix, then A^T
is _____ and (A^T)^−1 = ______

Answer 63

1. invertible
2. (A^−1)^T

Question 64

Theorem 2.4.4
(e) If A is invertible matrix then A^n is ____ for all integers n ≥ 0 and (A^n)^−1 = ______

Answer 64

1. invertible
2. (A^−1)^n

Question 65

Definition If A is invertible and n ≥ 0 an integer we define A^−n = ____ = ____

Answer 65

(A^−1)^n = (A^n)^−1

Question 66

Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is _______;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.

Answer 66

invertible

Question 67

Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution _____;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.

Answer 67

x = 0

Question 68

Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is ____;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.

Answer 68

I_n

Question 69

Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has __ ____ _____ for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.

Answer 69

a unique solution

Question 70

Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = ____

Answer 70

I_n.

Question 71

A matrix transformation is invertible

Answer 71

if there exists a matrix transformation T’: Rn → Rn with T’ ◦ T = I_n and T ◦ T’ = I_n

Question 72

If the matrix transformation T has an inverse, then its matrix A must be

Answer 72

invertible

Question 73

The geometric view of the inverse of a transformation provides a way to find the inverse of the matrix A. For example if the matrix A induces a reflection about the x-axis in R2, then A^−1 can be found by …

Answer 73

finding the transformation that reverses that reflection

Question 74

Elementary matrix

Answer 74

an n × n matrix which can be obtained by performing one elementary row operation on
an identity matrix In. We say that E is of type I, II or III if the operation is of that type.

Question 75

Performing elementary row operations is entirely equivalent to

Answer 75

multiplying by elementary matrices

Question 76

Lemma 2.5.1 Let E be the elementary matrix obtained by performing an elementary row operation on Im. If the same elementary row operation is performed on an m × n matrix A, then the result is the matrix ___

Answer 76

EA.

Question 77

Lemma 2.5.2 Each elementary matrix is _____, and its inverse is _______.

Answer 77

1. invertible
2. an elementary matrix of the same type

Question 78

Theorem 2.5.1 Suppose A is an m × n matrix and A → B by
elementary row operations. Then,
a) B =______
b) U can be computed by [A | Im] –ERO’s–> [B | U] using the operations carrying A → B.
c) U = E_kE__k−1. . . E_2E_1 where E_1, E_2, . . . , E_k are the elementary matrices corresponding (in order) to the elementary row operations carrying A to B.

Answer 78

UA where U is an m × m invertible matrix .

Question 79

Theorem 2.5.1 Suppose A is an m × n matrix and A → B by
elementary row operations. Then,
a) B = UA where U is an m × m invertible matrix .
b) U can be computed by ______
c) U = E_kE__k−1. . . E_2E_1 where E_1, E_2, . . . , E_k are the elementary matrices corresponding (in order) to the elementary row operations carrying A to B.

Answer 79

[A | Im] –ERO’s–> [B | U] using the operations carrying A → B.

Question 80

Theorem 2.5.1 Suppose A is an m × n matrix and A → B by
elementary row operations. Then,
a) B = UA where U is an m × m invertible matrix .
b) U can be computed by [A | Im] –ERO’s–> [B | U] using the operations carrying A → B.
c) U = _____

Answer 80

E_kE__k−1. . . E_2E_1 where E_1, E_2, . . . , E_k are the elementary matrices corresponding (in order) to the elementary row operations carrying A to B.

Question 81

Theorem 2.5.2 A square matrix is invertible if and only if

Answer 81

it is a product of elementary matrices.

Question 82

Definition A linear transformationfrom Rn
to Rm is a mapping T : Rn → Rm such that for all x, y ∈ Rn and scalars c ∈ R,
a) _____;
b) T(cx) = cT(x).

Answer 82

T(x + y) = T(x) + T(y)

Question 83

Definition A linear transformationfrom Rn
to Rm is a mapping T : Rn → Rm such that for all x, y ∈ Rn and scalars c ∈ R,
a) T(x + y) = T(x) + T(y);
b) _____.

Answer 83

T(cx) = cT(x)

Question 84

Theorem 2.6.2 Let T : Rn → Rm be a linear transformation. Then T is a ______ induced by the unique m × n matrix A, where the jth column of A is T(ej and A =[ T(e1) T(e2) · · · T(en) ], where {e1, e2, . . . , en} is _____, that is the columns of the identity matrix I_n

Answer 84

1. matrix transformation
2. the standard basis for Rn