Determinants

Question 1

Determinant gives what

Answer 1

volume of cuboid

Question 2

Determinants applied on ___________

Answer 2

Determinants applied on square matrix

Question 3

Definition of determinant

Answer 3

A factor which determines whether the linear system of equations have unique solution or not

Question 4

Determinant Formula

Answer 4

|Aₙₓₙ|= Σ ⁿⱼ₌₁ (-1)¹⁺ʲ a₁ⱼ δ₁ⱼ
(fix i = 1 (understood as first row is fixed))
fix any row or column

where δᵢⱼ= minor of an element aᵢⱼ
= it is determinant of sub-matrix obtained by eliminating iᵗʰ row and jᵗʰ column of element aᵢⱼ

Question 5

δᵢⱼ what it is

Answer 5

δᵢⱼ= minor of an element aᵢⱼ
= it is determinant of sub-matrix obtained by eliminating iᵗʰ row and jᵗʰ column of element aᵢⱼ

Question 6

Minor of an element

Answer 6

δᵢⱼ= minor of an element aᵢⱼ
= it is determinant of sub-matrix obtained by eliminating iᵗʰ row and jᵗʰ column of element aᵢⱼ

Question 7

Number of terms in determinant expansion of A

Answer 7

Number of terms in determinant expansion of Aₙₓₙ = n!

Question 8

|A| edit this cardddddd

Answer 8

|Aᵀ|

Question 9

|AB|

Answer 9

|AB| = |A||B| = |B||A|=|BA|

Question 10

|Aⁿ|

Answer 10

|A|ⁿ

Question 11

|A⁻¹|

Answer 11

|A⁻¹| = |A|⁻¹ = 1/|A|

Question 12

What happens to the determinant of a 2x2 matrix when a scalar multiplies one row or one column?

Answer 12

Remain

Question 13

remainn

Answer 13

aaa

Question 14

|k Aₙₓₙ|

Answer 14

kⁿ |A|

Question 15

What is the result of multiplying the determinant of a matrix A by a scalar k? State the formula.

Answer 15

|k Aₙₓₙ|= kⁿ |A|

Question 16

|adj Aₙ|

Answer 16

|A|ⁿ⁻¹

Question 17

adj(adj Aₙ)

Answer 17

|A|ⁿ⁻² A

Question 18

|adj(adj Aₙ)|

Answer 18

remain

Question 19

Determinant of lower triangular matrix or upper triangular matrix

Answer 19

Product of its principal diagonal elements

Question 20

Determinant of diagonal matrix

Answer 20

Product of its principal diagonal elements

Question 21

Determinant of skew matrix

Answer 21

The determinant of skew matrix of odd order is zero and of even order is a perfect square

Question 22

Determinant of idempotent matrix

Answer 22

0 or 1

Question 23

Determinant of involuntary matrix

Answer 23

± 1

Question 24

Determinant of orthogonal matrix

Answer 24

± 1

Question 25

Determinant of unitary matrix

Answer 25

± 1

Question 26

Relationship between linear dependency/linear independency and determinant

Answer 26

If any two row or column are linear dependent then the determinant of matrix is zero

Question 27

Determinant in Rᵢ <—> Rⱼ elementary transformation

Answer 27

|B| = - |A|

Question 28

Determinant in Rᵢ —> kRⱼ elementary transformation

Answer 28

|B| = k|A|

Question 29

Determinant in Rᵢ —> Rᵢ + cRⱼ elementary transformation

Answer 29

|B| = |A|

Question 30

Determinant in Cᵢ <—> Cⱼ elementary transformation

Answer 30

|B| = - |A|

Question 31

Determinant in Cᵢ —> CRⱼ elementary transformation

Answer 31

|B| = k|A|

Question 32

Determinant in Cᵢ —> Cᵢ + CRⱼ elementary transformation

Answer 32

|B| = |A|