Chapter 1.3 - 1.4

Question 1

size of a matrix

Answer 1

rows X columns

Question 2

What is a scalar?

Answer 2

constant

Question 3

How do you multiply two matrices?

Answer 3

Question 4

How do you TRANSPOSE (A^T ) matrix?

Answer 4

interchange rows and columns

Question 5

What is the trace of a matrix?

Answer 5

sum of the entries on the main diagonal of a SQUARE matrix

Question 6

What property do all matrix traces have?

Answer 6

Commutative property: trace AB = trace BA

Question 7

If you know that AB + BA is defined, then A and B are ____

Answer 7

square matrices

Question 8

Matrix Properties:

A + B =

Answer 8

A + B = B + A

Question 9

Matrix Properties:

A + (B + C) =

Answer 9

A + (B + C) = (A + B) + C

Question 10

Matrix Properties

Associative law for multiplication:

A(BC) =

Answer 10

A (BC) = (AB) C

Question 11

Matrix Properties

Left distributive law:

A (B + C) =

Answer 11

A (B + C) = AB + AC

Question 12

Matrix Properties

Right distribution law:

(B + C) A =

Answer 12

(B + C) A = BA + CA

Question 13

Properties DON’T WORK for Matrices

Not commutative:

AB ≠

Answer 13

AB ≠ BA

Question 14

Properties DON’T WORK for Matrices

If AB = AC, then does B always = C?

Answer 14

NO, if AB = AC

B ≠ C

Question 15

If AB = 0, then does A or B have to = 0?

Answer 15

NO: If AB = 0 then A or B don’t necessarily have to be = 0

Question 16

What does it mean to say that two matrices COMMUTE?

Answer 16

AB = BA

Question 17

What is an IDENTITY matrix?

Answer 17

square matrix with 1s on the diagonal and 0s everywhere else

Question 18

A I = ?

Answer 18

A I = A

Question 19

Can an identity matrix have an all 0 row?

Answer 19

NO

Question 20

What is an INVERTIBLE matrix?

Answer 20

If AB = BA = I

Question 21

B is called the inverse of A if ___

Answer 21

B = A^-1

Question 22

If A and B are invertible then ____

Answer 22

B = A^-1 and A = B^-1

Question 23

What does it mean if a matrix is SINGULAR?

Answer 23

Not invertible

Question 24

Are matrix inverses unique?

Answer 24

YES

Question 25

if B and C are both inverses of matrix A, then \_\_\_\_\_

Answer 25

B = C

Question 26

A A^-1 = A^-1A =

Answer 26

A A^-1 = *I* A^-1A = *I*

Question 27

How do you determine if a 2x2 matrix is invertible?

Answer 27

Question 28

Solution of a linear system by matrix inversion:

Answer 28

Question 29

(A^T)^T = (A + B)^T = (kA)^T = (AB)^T =

Answer 29

(A^T)^T = A (A + B)^T = A^T + B^T (kA)^T = kA^T (AB)^T = A^T B^T

Question 30

if A and B are invertible matrices with the same size, then \_\_\_

Answer 30

AB is invertible and (AB)^-1 = B^-1 A^-1

Question 31

If A is invertible, then A^T is also invertible then \_\_\_

Answer 31

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

Question 32

A⁰ =

Answer 32

A⁰ ​ = *I*

Question 33

If A is invertible, A^-n =

Answer 33

A^-n = (Aⁿ)^-1

Question 34

Do the laws of exponents still work for matrices?

Answer 34

YES

Question 35

Does (A+B)² = A² + 2AB + B²

Answer 35

(A+B)² ≠ A² + 2AB + B² Because AB ≠ BA

Question 36

Is a square matrix a row/column of 0s invertible?

Answer 36

NOT invertible → because no matrix B can be found such that BA=I