2.1/2.2

Question 1

What does A_32 mean?

Answer 1

3rd row 2nd column

Question 2

what are diagonal entries?

Answer 2

if [1 2 3; 4 5 6; 7 8 9;] then 1 5 and 9 are the diagonal entry

Question 3

do the matrices have to be the same size to do addition?

Answer 3

yes

Question 4

what are the properties of matrix addition?

Answer 4

a. A + B = B + A
b. (A+B)+C=A+(B+C)
c. A + 0 =A
d.r(A+B)=rA+rB
e.(r+s)A=rA+sA
f.r(sA)=(rs)A

Question 5

If A is mxn matrix, B is nxp and x is in R^p, denote the columns of B by b_1,…,b_p and the entries in x and x_1,…,x_p. Then,

Answer 5

Bx=x_1 b_1+…+x_p b_p

By linearity of multiplication by A…
A(Bx) =A(x_1 b_1)+…+A(x_p b_p)
=x_1 Ab_1+…+x_p Ab_p

Question 6

The vector A(Bx) is a linear combination of the vector Ab_1,…,Ab_1 using the entries in x as weights. This linear combination in matrix notation is:

Answer 6

A(Bx)=[Ab_1 ….. Ab_p]x
Transforms x into A(Bx)

Question 7

If A is an m x n matrix and if the B is an n x p matrix w/columns are Ab_1 … Ab_p. That is… (What is the definition of matrix multiplication?)

Answer 7

AB = A[b_1 …. b_p] = [Ab_1 …. Ab_p]

Question 8

whats the row column rule?

Answer 8

when you multiply matrices, write the sizes next to each other, then if the middle of the sizes are the same number, you cross them out then your size of the multiplied matrix is the two numbers remaining

Question 9

is matrix multiplication communicative?

Answer 9

no unless your using the identity matrix

Question 10

what is is A^k intuitively?

Answer 10

list of k matrices(A)

Question 11

what is A^T?

Answer 11

the rows become the columns and the columns become the rows.

Question 12

thrm abt transposing matrices?

Answer 12

a. (A^T)^T = A
b. (A+B)^T = A^T + B^T
c.for any scalar, r, (rA)^T = rA^T
d. (AB)^T = B^T*A^T

Question 13

what’s a singular and non singular matrix?

Answer 13

singular matrix- a square matrix that has no inverse
nonsingular matrix-an invertible matrix

Question 14

what is the det A?

Answer 14

if A = [a b; c d;]. det A = ad-bc

Question 15

what is the general formula for the inverse of a 2 x 2 matrix?

Answer 15

if A = [a b; c d;], then A^-1 is
1/(ad-bc) * [d -b; -c a;]

Question 16

how do we use the inverse of a 2x2 matrix A to solve a linear system?

Answer 16

if Ax=b then there is unique sol-n x=A^-1 * b.

Question 17

how to find the inverse of an n x n matrix where n > 2?

Answer 17

[A I] = [I A^-1]

Attach the identity matrix (I) to A. find the rref to get [I A^-1]. The identity matrix should be on the left now and what’s on the right is A^-1

Question 18

T or F:
A product of invertible n x n matrices is​ invertible, and the inverse of the product is the product of their inverses in the same order.

Answer 18

The statement is false. If A and B are invertible​ matrices, then (AB)^-1 = B^-1 * A^-1.

Question 19

T or F: In order for a matrix B to be the inverse of​ A, both equations AB = I and BA = I must be true.

Answer 19

The statement is true. The product of a matrix and its inverse is the identity matrix.

Question 20

can only square matrices have inverses?

Answer 20

Yes, only square matrices can have inverses.

Question 21

T or F: If A =​ [ a b; c d;], and ab - cd ≠ 0, then A is invertible.

Answer 21

false. it’s if ad-bc ≠ 0 then A is invertible

Question 22

how do you check if a matrix is the inverse of another?

Answer 22

check if they equal I

Question 23

what is an elementary matrix?

Answer 23

a matrix that was the identity matrix but had one elementary row operation(scale, swap, replace)