Chapter 2

Question 1

Zero matrix

Answer 1

a m x n matrix of zeros denoted 0

Question 2

triangular matrix

Answer 2

n x n matrix where entries above or below the main diagonal are zero

Question 3

diagonal matrix

Answer 3

n x n matrix where all entries off the main diagonal are zero

Question 4

identity matrix

Answer 4

n x n diagonal matrix with 1’s along the main diagonal, denoted I_n

Question 5

When are two matrices equal?

Answer 5

A = B when A and B have the same size and corresponding entries

Question 6

How are two matrices added?

Answer 6

(A + B): add entry-wise (must have the same size)

Question 7

How is a matrix subjected to scalar multiplication?

Answer 7

(kA) scale each entry by k
-A = (-1)A
A-B = A + (-B)

Question 8

How is a matrix transposed?

Answer 8

A^t is the matrix whose columns are the rows of A

ij entry of A = ji entry of A^t

Question 9

How is matrix multiplication performed?

Answer 9

If A is an m x n matrix and B is an n x p matrix with columns b1, … , bp, then
AB = A[b1 b2 … bp] = [Ab1 Ab2 … Abp]
(Across and Down rule:
[AB]ij = (row i of A)*(col j of B)

Question 10

What are the properties of matrix addition?

Answer 10

Let A, B, and C be matrices with sizes such that the indicated operations are defined. The following hold:

i) A + B = B + A
ii) A + ( B + C ) = ( A + B ) + C
iii) A + 0 = A

Question 11

What are the properties of scalar multiplication of matrices?

Answer 11

Let A, B, and C be matrices with sizes such that the indicated properties are defined. Let r, s be scalars. The following hold:

iv) r(A + B) = rA + rB
v) (r + s)A = rA + sA
vi) r(sA) = (rs)A

Question 12

What are the properties of matrix multiplication?

Answer 12

Let A, B, and C be matrices with sizes such that the indicated operations are defined. Let r, s be scalars. The following hold:

vii) A(BC) = (AB)C
viii) A(B + C) = AB + AC
ix) (A + B)C = AC + BC
x) r(AB) = (rA)B = A(rB)
xi) I_m * A = A = A* I_m (where I_m represents the m x m identity matrix)

Question 13

What are the properties of matrix transposition?

Answer 13

Let A, B, and C be matrices with sizes such that the indicated operations are defined. Let r, s be scalars. The following hold:

xii) (A^T)^T = A
xiii) (A + B)^T = A^T + B^T
xiv) (r*A)^T = r * A^T
xv) (AB)^T = (B^T)(A^T)

Question 14

There are some possible outcomes in matrix arithmetic that don’t occur in usual scalar arithmetic. What are they?

Answer 14

The following are possible for matrices A, B, and C:

1: AB is not equal to BA
2: AB = AC but B is not equal to C
3: AB = 0 but A and B are not equal to zero

Question 15

When is a matrix B the inverse of a matrix A?

Answer 15

If A is a square matrix, then B is called the inverse of A if AB = BA = I
Additionally, the following phrases are equivalent: has an inverse, invertible, nonsingular
If A has an inverse, then it is unique and is denoted A^-1

Question 16

What are the properties of inverses? (Theorem 2.6)

Answer 16

If A and B are invertible n x n matrices, then:

1: (A^-1)^-1 = A
2: (AB)^-1 = (B^-1)(A^-1) (socks-shoes property)
3: (A^T)^-1 = (A^-1)^T

Question 17

Theorem 2.5

Answer 17

If the n x n matrix A is invertible, then the system Ax = b has a unique solution for each b in R^n, namely x = (A^-1)b

Question 18

How can the inverse of a 2 x 2 matrix be found? (Theorem 2.4)

Answer 18

Let A = [ a, b; c, d]. Then A is invertible if and only if ad-bc is not equal to 0. If A is invertible, then
A^-1 = (1/(ad-bc))*[d, -b; -c, a]
(the number ad-bc is called the determinant of A)

Question 19

How can the inverse of a matrix be found in general?

Answer 19

An n x n matrix A is invertible if and only if A is row equivalent to I, and in this case, any sequence of elementary row operations that reduces A to I, also transforms I into A^-1
In other words, the theorem says:
A is invertible if and only if [ A | I ] is row equivalent to [ I | A^-1 ]

Question 20

Elementary matrix

Answer 20

One obtained by performing a single elementary row operation on I

Question 21

Facts about elementary matrices

Answer 21

1: if E is an elementary matrix, then EA is the matrix that results if the elementary row operation that produced E is performed on A. (We can perform row operations via matrix multiplication)
2: each elementary matrix is invertible, and E and E^-1 represent “inverse operations”

Question 22

What is the endpoint of LU factorization?

Answer 22

Factoring the m x n matrix A into the matrix product LU where:

1: L is a lower triangular m x m matrix with 1’s on the main diagonal
2: U is an RE form of A

Question 23

What is the algorithm for LU factorization?

Answer 23

1: Reduce A to RE form U using ONLY (lower) row replacement operations (no scaling or interchanging!) if possible.
2: Keep track of each replacement operation
3: Form L by applying the inverse operations, in reverse order, to the identity I

Question 24

Why do we insist on only using certain row replacement operations in our row reduction?

Answer 24

All other row operations mess up L as a unit lower triangular matrix

Question 25

Does every matrix A have an LU factorization?

Answer 25

Nah

Question 26

Why is LU factorization useful?

Answer 26

If solving the system Ax = b for repeated values of b, the LU solution method involves fewer individual addition/multiplication computations than multiplication by A^-1 (if it exists) or repeated elementary row reduction.