Special type of SQ matrices

Question 1

Entries a_ij, i = j, are called?

Answer 1

Main diagonal entries

Question 2

The trace of A, tr(A) is?

Answer 2

The sum of diagonal entries of A

Question 3

Properties of trace

Answer 3

tr(A+B) = tr(A) + tr(B)
tr(AB) = tr(BA)

Question 4

If a_ij = 0 when i is NOT = j is called?

Answer 4

Diagonal (entries OUTSIDE the main diagonal are 0)

Question 5

A diagonal matrix whose diagonal entries are = is called?

Answer 5

Scalar matrix

Question 6

Is 0 matrix a scalar or diagonal matrix?

Answer 6

Both

Question 7

What type of matrix is an identity matrix

Answer 7

Scalar

Question 8

An nxn scalar matrix whose diagonal entries are all = to 1

Answer 8

Identity matrix

Question 9

If a_ij = 0 when i > j

Answer 9

Upper triangular (all entries BELOW main diagonal = 0)

Question 10

If a_ij = 0 when i < j

Answer 10

Lower triangular (all entries ABOVE main diagonal = 0)

Question 11

All diagonal, scalar, and identity are upper or lower triangular?

Answer 11

Both upper and lower triangular

Question 12

If A^T = A

Answer 12

Symmetric

Question 13

If A^T = -A

Answer 13

Skew symmetric

Question 14

Requirements for skew and symmetric?

Answer 14

Must be a square matrix ksi ur transposing them so dpat same row & cols

Question 15

If A is skew, main diagonal entries of A are?

Answer 15

0
cuz, a_ij = -a_ij, and only 0 satisfies this
0 = -0
1 is NOT = to -1

Question 16

If A is a sq matrix, it can be expressed as A = S + K

Answer 16

True, this decomposition is unique

Question 17

Skew and sym formula

Answer 17

S = 1/2(A + A^T)
K = 1/2(A - A^T)

Question 18

Properties of square matrix A (exponents)

Answer 18

1.) A^3 = AxAxA
2.) A^p x A^q = A^p+q
3.) (A^p)^q = A^pq

Question 19

Does 0 property hold for matrix?

Answer 19

No, if AB = O, it doesn’t mean tht either A = O or B = O

Question 20

Does cancellation hold for matrix?

Answer 20

No, if AB = AC, it doesn’t mean B = C

Question 21

An nxn A is ____ if there exists an nxn B such tht AB = BA = In

Answer 21

Non singular/Invertable (take note: A n B must be square matrices)

Question 22

What if there’s no B tht can satisfy AB=BA=In?

Answer 22

A is singular/non invertable

Question 23

Any square matrix O is singular/noninvertable or nonsingular/invertable

Answer 23

Singular/noninvertable

Cuz there’s no matrix such tht if u mult to O will = to Identity, it will always = to O

Question 24

Properties of inverse

Answer 24

-if it exists, it’s unique
-if A&B r nonsingular, then AB is nonsingular
-refer to ntbk for 3-5

Question 25

Ref rules

Answer 25

1.) all rows containing entirely of 0s IF ANY, should be at the bottom of matrix 2.) 1st non 0 entry of each row (leading entry), tht doesn’t consist entirely of 0s is 1 3.) each leading entry is in a col to the RIGHT of the leading entry in the prev row

Question 26

For it to be an REF, bottom row should all be 0s

Answer 26

False

Question 27

RREF rules

Answer 27

If a col contains a leading entry of some row, then all other entries in the col are 0

Question 28

All rows need a leading entry to be an REF

Answer 28

True (excluding bottom sometimes)

Question 29

Gaussian elimination

Answer 29

Apply EROs to obtain REF

Question 30

Gauss jordan

Answer 30

Apply EROs to obtain RREF

Question 31

(REF/RREF) is unique

Answer 31

RREF

Question 32

How to verify inverse of a matrix?

Answer 32

Mult the og to its inverse, prod should be its identity