Chapter 2- Matrix Algebra

Question 1

Properties of Matrix Multiplication (Theorem 2)

Answer 1

1) A(BC) = (AB)C
2) A(B + C) = AB + AC
3) (B + C)A = BA + CA
4) r(AB) = (rA)B = A(rB) r is scalar
5) ImA = A = AIn
6) A^k = AAA … *A (k times)

Question 1

Warnings about multiplying matrices

Answer 1

1) In general, AB =/= BA
2) Cancellation laws do NOT hold for matrices. That is, if AB = AC, then in general, B =/= C.
3) If AB = 0, that does NOT imply A = 0 or B = 0

Question 2

Definition of transposing a matrix A

Answer 2

A is an mxn matrix, then the transpose of A, denoted by A^T, whose switching the rows and columns to an nxm matrix

Question 3

What are the properties of transposing a matrix?

Answer 3

Let A and B denote matrices whose sizes are appropriate for the following sums and products.

1) (A^T)^T = A
2) (A + B)^T = A^T + B^T
3) for any scalar r, (rA)^T = rA^T
4) (AB)^T = B^T*A^T

The transpose of a product of matrices equals the product of their transposes in the reverse order

Question 4

If A and B are matrices, how do we know if AB is defined?

Answer 4

AB is defined iff the number of columns for A is equal to the number of rows for B

Question 5

How to compute AB separately using its components

Answer 5

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B (B has p columns).

AB = A[b1 b2 b3 … bp]

Question 6

Let A and B be matrices. If A and B commute with one another…

Answer 6

… then AB = BA. Also, they are both square matrices.

Question 7

If a matrix A is 5x3 and the product AB is 5x7, what is the size of B?

Answer 7

3x7

Question 8

True or false. Why?

If A and B are 2x2 matrices with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2]

Answer 8

False

AB = A[b1 b2] = [Ab1 Ab2]

Question 9

True or false. Why?

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.

Answer 9

True

AB = [Ab1 Ab2 … Abp]

Question 10

True or false. Why?

The transpose of a product of matrices equals the product of their transposes in the same order.

Answer 10

False

(AB)^T = B^T*A^T
Says the property of transposing matrices

Question 11

True or false. Why?

If A is an nxn matrix,
then (A^2)^T = (A^T)^2

Answer 11

True

(A^2)^T = (AA)^T = A^TA^T
= (A^T)^2

Question 12

Suppose the third column of B is the sum of the first two columns. What can be said about the third column of AB? Why?

Answer 12

Third column of AB is the sum of the first two columns of AB

Third column AB = Ab3 = A(b1 + b2)
= Ab1 + Ab2

By a property of matrix-vector multiplication

Question 13

Suppose the first two columns, b1 and b2, of B are equal, what can we say about AB. Why?

Answer 13

Those columns will be equal to each other

AB = [Ab1 Ab2 … ]

If b1 and b2 are equal, the first two columns of AB are interchangeable.

Question 13

Suppose the last column of AB is entirely zeros but B itself has no column of zeros. What can be said about the columns of A? Why?

Answer 13

Columns of AB are linearly dependent. Figure out why

Question 13

True or false. Why?

If A and B are nxn and invertible, then A^(-1)B^(-1) is the inverse of AB

Answer 13

False.

The product of nxn invertible matrices is invertible, and the inverse is the product of their inverses in the REVERSE order.

States theorem 6

Question 14

If A is a 2x2 matrix, what is the determinant of A and what can that tell us?

Answer 14

Det(A) = ad - bc

Det(A) =/= 0, then A is invertible
Det(A) = 0, then A is NOT invertible

Question 15

What is a singular matrix?

What is a nonsingular matrix?

Answer 15

Singular matrix: matrix that is NOT invertible

Nonsingular matrix: matrix that is invertible

Question 15

Theorem 5?

If A is an invertible matrix, what can we say about Ax = b?

Answer 15

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

Question 16

Theorem 6

Facts about invertible matrices

Answer 16

1) (A^(-1))^(-1) = A

2) (AB)^(-1) = B^(-1)A^(-1)
The inverse of AB is the product of the inverses of A and B in the reverse order.

3) (A^(T))^(-1) = (A^(-1))^(T)

Question 16

Theorem 7

Best way to visualize an invertible matrix. So pay attention!

Answer 16

An nxn matrix A is invertible iff A is row equivalent to I, and in this case, and sequence of elementary row operations that reduces A to I also transforms I to A^(-1)

Question 17

Algorithm for finding A^(-1)

Answer 17

If we place A and I side-by-side to form an augmented matrix

[A I], then row operations to transform A to I will also transform I to A^(-1)

Question 18

True or false. Why?

If A is invertible, then the inverse of
A^(-1) is A itself

Answer 18

True

Question 24

What is the invertible matrix theorem? (IMT)

Answer 24

Let A be a square nxn matrix. These statements are either all true or all false.

1) A is an invertible matrix
2) A is row equivalent to I (n pivot positions)
3) Ax = b has only one solution for each b in R^n (existence and uniqueness)
4) Columns of A span R^n
5) A^T is invertible

Question 25

True or false. Why?

If A is invertible, then the columns of A are linearly dependent.

Answer 25

False

Remember the algorithm for finding A-inverse. Must reduce A to rref with pivots in every row and column

Question 25

What is an upper triangular matrix?

Lower triangular matrix?

Answer 25

Upper: Matrix whose entries below the main diagonal are all zero’s

Lower: entries above main diagonal are all zero’s

Question 26

Can a square matrix with two identical columns be invertible? Explain

Answer 26

No because those columns are linearly dependent, violation IMT

Question 27

What is an upper triangular matrix?

Lower triangular matrix?

Answer 27

Upper: Matrix whose entries below the main diagonal are all zero’s

Lower: entries above main diagonal are all zero’s

Question 28

Can a square matrix with two identical columns be invertible? Explain

Answer 28

No because those columns are linearly dependent, violation IMT