Matrix

Question 1

Trace(A)

Answer 1

sum of principal diagonal elements

Question 2

Orthogonal matrix

Answer 2

A⁻¹ = Aᵀ
or AAᵀ = AᵀA = I

Question 3

Matrix multiplication is commutative or not
What it means

Answer 3

Non commutative
AB != BA

Question 4

Symmetric matrix

Answer 4

Aᵀ = A

Question 5

(Aᵀ)ⁿ

Answer 5

(Aⁿ)ᵀ

Question 6

If A is idempotent matrix then what it tells about Aⁿ

Answer 6

then Aⁿ = A

Question 7

How to identify matrix is symmetric if it is given

Answer 7

The elements of upper triangular matrix and lower triangular matrix are symmetric about principal diagonal
Diagonal elements of matrix are real

Question 8

What is index or degree k of Nilpotent matrix

Answer 8

How many times it is required to multiply to get the null matrix is understand as degree of it

Question 9

Hermitian matrix

Answer 9

Ā = Aᵀ (A bar = A transpose)
or
(Ā)ᵀ = A
or
Aθ = A (A raise to the power theta = A)

Question 10

A is symmetric, what it tells about Aⁿ

Answer 10

It is also symmetric

Question 11

If AB = B and BA = A then

Answer 11

A and B both are idempotent matrices

Question 12

Trace(A-B)

Answer 12

Trace(A)-Trace(B)

Question 13

Conjugate of a matrix

Answer 13

it changes the sign of imaginary part of the complex element

Question 14

How to represent conjugate matrix

Answer 14

Ā (A bar or A with macron)

Question 15

Every square matrix can be written as
1.) _______________________
2.) _______________________

Answer 15

1.) Sum of symmetric and skew symmetric matrices
Anxn = 1/2(A + Aᵀ) +1/2(A-Aᵀ)
2.) Sum of hermitian and skew-hermitian matrix
Anxn = 1/2(A + Aθ) + 1/2(A - Aθ).
(here θ is in power)

Question 16

complex matrix

Answer 16

A matrix having at least one element complex

Question 17

Skew Hermitian matrix

Answer 17

Ā = -Aᵀ (A bar =- A transpose)
or
(Ā)ᵀ = -A
or
Aθ = -A (A raise to the power theta = -A)

Question 18

Diagonal elements of symmetric matrix

Answer 18

Real

Question 19

How to identify matrix is skew symmetric matrix if it is given

Answer 19

Diagonal elements are zero
Lower triangular elements = -Upper triangular elements

Question 20

Involuntary matrix

Answer 20

A² = I
or
A = A⁻¹

Question 21

Idempotent matrix

Answer 21

A² = A

Question 22

Trace(k(A))

Answer 22

k(Trace(A))

Question 23

Trace(A+B)

Answer 23

Trace(A)+Trace(B)

Question 24

Matrix multiplication is associative or not
What it means

Answer 24

Associative
A(BC) = (AB)C

Question 25

(AB)ᵀ

Answer 25

BᵀAᵀ

Question 26

All symmetric matrix are ___________ matrix

Answer 26

Hermitian

Question 27

If any matrix is multiplied for numbers of time so that the result is null matrix is called ________________

Answer 27

Nilpotent matrix

Question 28

A is skew-symmetric matrix, what it tells about Aⁿ

Answer 28

Aⁿ is symmetric if n is even Aⁿ is skew-symmetric if n is odd

Question 29

Diagonal elements of Hermitian matrix

Answer 29

if elements of diagonal are real, then there is no affect on diagonal elements

Question 30

A matrix is multiplied by itself and you are getting identity matrix then

Answer 30

matrix and its inverse both are equal and matrix is involuntary matrix

Question 31

Diagonal elements of Skew symmetric matrix

Answer 31

0

Question 32

Skew symmetric matrix

Answer 32

A = -Aᵀ or A + Aᵀ=0

Question 33

If a matrix is multiplied by itself and you are getting same matrix then such matrix is _________________

Answer 33

Idempotent matrix

Question 34

Trace(Aᵀ)

Answer 34

Trace(A)

Question 35

(Aᵀ)ᵀ

Answer 35

A

Question 36

(A+B)ᵀ

Answer 36

Aᵀ+Bᵀ

Question 37

order of Aᵀ

Answer 37

nxm

Question 38

Nilpotent matrix

Answer 38

Aᵏ = Onxn (Null matrix) here k is index or degree

Question 39

For Trace(A), which condition is required

Answer 39

The matrix should be square

Question 40

If AB = A and BA = B then

Answer 40

A and B both are idempotent matrices

Question 41

Conjugate matrix of real elements

Answer 41

No affect on matrix

Question 42

Hint of Hermitian matrix

Answer 42

tranjugate = transpose + conjugate

Question 43

(-A)ᵀ

Answer 43

-Aᵀ

Question 44

Unitary matrix

Answer 44

AAθ = Aθ A = I or Aθ = A⁻¹ (here θ is superscript)

Question 45

Singular matrix

Answer 45

|A| = 0

Question 46

Non-singular matrix

Answer 46

|A| ≠ 0

Question 47

Inverse of matrix

Answer 47

A⁻¹ = adjA / |A|, where |A| != 0

Question 48

(A+B)²

Answer 48

(A+B)² = A² + AB + BA + B²

Question 49

The minimum number of multiplications required to multiply Aₘₓₙ with Bₙₓₚ

Answer 49

The minimum number of multiplications required to multiply Aₘₓₙ with Bₙₓₚ is mnp

Question 50

Matrices of same dimension means

Answer 50

same order i.e. if you have A of order mxn then order of B will be same i.e, mxn

Question 51

If A and D are similar matrices, what about their determinants

Answer 51

|A|=|D|

Question 52

(AB)ⁿ

Answer 52

BⁿAⁿ

Question 53

(AB)⁻¹

Answer 53

B⁻¹A⁻¹

Question 54

AAᵀ matrix

Answer 54

AAᵀ matrix symmetric matrix

Question 55

note about product of two symmetric matrices

Answer 55

Product of 2 symmetric matrices is need no to be symmetric unless A = B

Question 56

note about product of two skew symmetric matrices

Answer 56

Product of 2 skew-symmetric matrices is need no to be skew symmetric unless AB = -BA

Question 57

Geometrical definition of orthogonal matrix

Answer 57

A = [x₁ x₂ x₃ ... xₙ]ₙₓₙ is a orthogonal matrix whose vectors are orthonormalisation vectors. [Here x₁,x₂,x₃, ... xₙ are n tuple vectors i.e. they are n-dimensional column vectors i.e. have n components] OR A square matrix whose vectors are orthonormal vectors called orthogonal matrix.

Question 58

Partition matrices

Answer 58

Divide the matrix into number of blocks

Question 59

Type of partition matrices also their determinants and condition for it also

Answer 59

Question 60

Inverse of matrix (Proper formula)

Answer 60

Question 61

Inverterse of 2x2 matrix (shortcut)

Answer 61