Test 1 prep

Question 1

3 types of possible solutions:

Answer 1

1 ) Unique solution
2 ) Infinite solution
3 ) No solution

Question 2

Unique and infinite solutions are ______ solutions

Answer 2

consistent

Question 3

No solution is _________

Answer 3

inconsistent

Question 4

Difference between REF and RREF:

Answer 4

In RREF, the entries aside from leading ones have to be zeros whereas in REF, we only need leading ones as the first entry of a row

Question 5

True or false: a matrix can have multiple REFs, but only a unique RREF.

Answer 5

True

Question 6

“Number of leading variables of a matrix is called the _______ of that matrix”

Answer 6

rank (represented as IR^n)

Question 7

In the parametric solutions, there are 2 types of variables:

Answer 7

1 ) Leading variables
2 ) Free variables

Question 8

Only a square n x n matrix will have a main _______

Answer 8

diagonal

Question 9

Entries of a matrix are represented as:

Answer 9

a subscript ij

Question 10

In a subscript ij, what does ij represent:

Answer 10

i = row
j = column

Question 11

Matrix product can only be achieved if:

Answer 11

nº of columns in A is identical to nº of rows in B

Question 12

What is the proper way of writing a system if linear equations in matrix form?

Answer 12

AX = b

Question 13

What is a matrix transpose?

Answer 13

The interchange of rows and columns

Question 14

What are the 3 properties of AA^T or A^TA?

Answer 14

1 ) products are always square matrices
2 ) products are always symmetrical matrices
3 ) entries of the main diagonal are always positive

Question 15

What is the trace of a matrix?

Answer 15

Sum of main diagonal entries

Question 16

tr(A+B) =

Answer 16

tr(A) + tr(B)

Question 17

True or false: tr(A) = tr(A^T)

Answer 17

True

Question 18

tr(k*A) =

Answer 18

k*tr(A)

Question 19

tr(AB) =

Answer 19

tr(BA)

Question 20

A + B =

Answer 20

B + A

Question 21

(A + B) + C =

Answer 21

A + (B + C)

Question 22

In general, AB does not =

Answer 22

BA

Question 23

(AB)C =

Answer 23

A(BC)

Question 24

A(B + C) =

Answer 24

AB + AC (right distribution)

Question 25

(A + B)C =

Answer 25

AC + BC (left distribution)

Question 26

k(AB) =

Answer 26

(k*A)B = A(k*B)

Question 27

(a + b)C =

Answer 27

aC + bC

Question 28

(ab) C =

Answer 28

b(aC) = a(bC)

Question 29

What is the identity matrix?

Answer 29

square matrix (size depending on the other matrices in the equation) where all its entries are 0 and the main diagonal entries are 1

Question 30

A*A^-1 =

Answer 30

I

Question 31

How do you find determinant of 2x2 matrix? det(a) or IAI

Answer 31

ad–bc

Question 32

How do you find inverse of 2x2 matrix?

Answer 32

A^ -1 = 1/det(A) [exchange a with d; exchange signs of b and c]

Question 33

A^n =

Answer 33

A multiplied n number of times

Question 34

A^0 =

Answer 34

I subscript matrix size

Question 35

A^n * A^r =

Answer 35

A^(n+r)

Question 36

(A^n)^r =

Answer 36

A ^ (n*r)

Question 37

A^ -n =

Answer 37

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

Question 38

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

Answer 38

A

Question 39

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

Answer 39

(A^n)^-1 (A^-1)^T

Question 40

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

Answer 40

(A^-1 + B^-1) (A^T + B^T)

Question 41

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

Answer 41

k^-1 * A^-1 = (1/k) * A^-1 k * A^T

Question 42

(ABCD)^-1 = (ABCD)^T =

Answer 42

D^-1 * C^-1 * B^-1 * A^-1 D^T * C^T * B^T * A^T

Question 43

How do you find A in AX=b?

Answer 43

X=A^-1 * b

Question 44

What is a homogenous system?

Answer 44

solutions are = 0

Question 45

What is a diagonal matrix?

Answer 45

All entries beyond the main diagonal are 0 in a square matrix

Question 46

What are the 2 properties of the diagonal matrix?

Answer 46

1 ) the diagonal matrix is invertible if none of its main diagonal entries is 0 2 ) the power of the matrix can be distributed to the entries of the main diagonal

Question 47

What does the LOWER triangular matrix look like?

Answer 47

entries above main diagonal are 0

Question 48

What does the UPPER triangular matrix look like?

Answer 48

entries below main diagonal are 0

Question 49

If any entry of the main diagonal of the triangular matrix is 0:

Answer 49

it is SINGULAR

Question 50

SINGULAR =

Answer 50

NOT INVERTIBLE

Question 51

The main diagonal of a symmetric matrix acts as a _______

Answer 51

mirror line

Question 52

The transpose of the symmetrical matrix is _______ to original matrix

Answer 52

identical

Question 53

The inverse, if it exists, of the symmetrical matrix is _______ to original matrix

Answer 53

identical

Question 54

The inverse of a matrix only exists if:

Answer 54

det(A) does not = 0