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# Class 12-science RD SHARMA Solutions Maths Chapter 7 - Adjoint and Inverse of a Matrix

## Adjoint and Inverse of a Matrix Exercise Ex. 7.2

### Solution 3

Let

AA-1 = I

Applying R2 R2 - 2R1

Applying R2 R2/-5

Applying R1 R1 - 2R2

### Solution 17

Let

AA-1 = I

Applying R2 R2 - 2R1 and R3 R3 + 2R1

Applying R1 R1 - 2R2

Applying R1 R1 - R3 and R2 R2 - R3

Hence,

### Solution 18

Let

AA-1 = I

Applying R3 R1

Applying R2 R2 - 3R1 and R3 R3 - 2R1

Applying R2 -R2

Applying R1 R1 - R2 and R3 R3 + 5R2

Applying R3 -R3

Applying R2 R2 + 2R3

Hence,

## Adjoint and Inverse of a Matrix Exercise MCQ

### Solution 1

Correct option: (a)

|A-1| = |A|-1, (AT)-1 = (A-1)T, |A| 0 are properties of an invertible matrix.

### Solution 2

Correct option: (c)

### Solution 3

Correct option:(d)

### Solution 4

Correct option: (b)

### Solution 5

Correct option:(b)

If A is singular matrix then adjoint of A is also singular.

### Solution 6

Correct option: (a)

### Solution 7

Correct option: (c)

### Solution 8

Correct option:(a)

### Solution 9

Correct option: (c)

### Solution 10

Correct option: (c)

### Solution 11

Correct option: (d)

### Solution 12

Correct option: (d)

### Solution 13

Correct option: (a)

### Solution 14

Correct option: (b)

### Solution 15

Correct option: (c)

### Solution 16

Correct option: (d)

### Solution 17

Correct option: (d)

### Solution 18

Correct option: (b)

### Solution 19

Correct option: (d)

### Solution 20

Correct option: (c)

### Solution 21

Correct option: (c)

Adj A = |A| A-1, Det(A-1) = (det A)-1, (AB)-1 = B-1A-1 are all the properties of invertible matrix.

### Solution 22

Correct option: (b)

### Solution 23

Correct option: (a)

### Solution 24

Correct option: (b)

### Solution 26

Correct option: (d)

### Solution 27

Correct option: (b)

### Solution 28

Correct option: (d)

### Solution 29

Correct option: (b)

### Solution 30

Correct option: (a)

### Solution 31

Correct option: (a)

### Solution 32

Given: A and B are invertible matrices

Relation between inverse and adjoint of a matrix is given by

We know that, AA-1 = I

As

But

### Solution 33

A matrix is invertible or its inverse exists if determinant is 0.

Now,

Therefore, A-1 exists if |A| = 0

i.e. if

i.e. if

### Solution 34

Given: A2 = A

(I - A)3 + A = (I - A)2(I - A) + A

= (I2 - 2AI + A2)(I - A) + A

= (I- 2A + A)(I - A) + A … (Since A2 = A)

= (I - A)(I - A) + A

= I2 - 2AI + A2 + A

= I - 2A + A + A

= I

Hence, (I - A)3 + A = I.

### Solution 35

The matrix is non-invertible if its determinant is 0.

This matrix will not be invertible if

i.e. if

## Adjoint AND Inverse of a Matrix Exercise Ex. 7VSAQ

### Solution 29

Given:  which is in the form X = AB

Now, the column operation is applicable on X and B simultaneously when the equation is X = AB

Therefore, we'll apply the column operation on the first and third matrices.

Applying C2 C2 + 2C1

### Solution 30

Given:  which is in the form AB = X

Now, the row operation is applicable on A and X simultaneously when the equation is AB = X

Therefore, we'll apply the row operation on the first and third matrices.

Applying R2 R2 + R1

Given: |A| = 4