# HOW TO FIND A MATRIX FROM ITS ADJOINT

How to Find a Matrix From its Adjoint :

Here we are going to see some example problems finding adjoint of matrix.

## What is the Formula to Find a Matrix from its Adjoint

if A is a non-singular matrix of order 3, then we get ## How to Find a Matrix From its Adjoint - Practice Questions

Question 1 : |adj A|  =  2(24 - 0) + 4(-6 - 14) + 2(0 + 24)

=  2(24) + 4(-20) + 2(24)

=  48 - 80 + 48

=  96 - 80

=  16

√adj A  =  √16  =  4 By applying these values in the formula, we get ## Finding the Inverse of a Matrix Using the Adjoint

Question 2 :  |adj A|  =  0(12 + 0) + 2(36 - 18) + 0(0 + 6)

=  0 + 2(18) + 0

√adj A  =  √36  =  6 Question 3 : Solution : Question 4 : Solution : |A|  =  1 + tan2 x

|A|  =  sec2 x  Hence proved.

Question 4 : Solution : 5a - b  =  14  -------(1)

3a - 2b  =  7 -----(2)

(1)x2==> 10a - 2b  =  28

3a - 2b  =  7

(-)   (+)    (-)

-----------------

7a  =  21

a  =  21/7  =  3

By applying the value of a in (1), we get

5(3) - b  =  14

-b  =  14 - 15

-b  =  -1

b  =  1

5c - d  =  7  -------(3)

3c - 2d  =  7 -----(4)

(3) x 2  ==>   10c - 2d  =  14

3c - 2d  =  7

(-)   (+)     (-)

-----------------

7c  =  7

c  =  7/7  = 1

By applying the value of c in (3), we get

5(1) - d  =  7

5 - d  =  7

-d  =  7 - 5

d  =  -2 Question 5 : Solution : 3(a-c)+(b-d)  =  1

3a - 3c + b - d  =  1 ---(1)

-2(a-c)+(b-d)  =  1

-2a+2c + b - d  =  1------(2)

6a + 2b  =  2  ----(3)

-4a + 2b  =  2 -----(4)

(3)/2  ==>  3a + b  =  1

(4)/2  ==> -2a + b  =  1

By subtracting the above equations, we get

a  =  0

By applying the value of a in (3), we get

3(0) + b  =  1

b  =  1

From (1),

3a - 3c + b - d  =  1 ---(1)

-3c + 1 - d  =  1

-3c - d  =  1 - 1

-3c - d  =  0 --(A)

From (2),

-2a + 2c + b - d  =  1------(2)

2c + 1 - d  =  1

2c - d  =  1 - 1

2c - d  =  0--(B)

By solving (A) and (B), we get

(A) - (B)

-3c - 2c  =  0

c  =  0

By applying the value of c in (A), we get

d  =  0 Question 6 : Solution :

|A|  =  0(0-1) - 1(0-1) + 1(1-0)

=  0 + 1 + 1

|A|  =  2   Hence proved.

Question 7 :

Decrypt the received encoded message [2    -3]  [20     4] with the encryption matrix and the decryption matrix as its inverse, where the system of codes are described by the numbers 1-26 to the letters A− Z respectively, and the number 0 to a blank space.

Solution :  Hence the required word is "HELP". After having gone through the stuff given above, we hope that the students would have understood, "How to Find a Matrix From its Adjoint".

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