HOW TO IDENTIFY IF THE GIVEN MATRIX IS SINGULAR OR NONSINGULAR

A square matrix A is said to be singular if |A| = 0. A square matrix A is said to be non-singular if | A | ≠ 0.

Question 1 :

Identify the singular and non-singular matrices:

Solution :

In order to check if the given matrix is singular or non singular, we have to find the determinant of the given matrix.

Hence the matrix is singular matrix.

Solution :

It is not equal to zero. Hence it is non singular matrix.

Solution :

Since the given matrix is skew matrix, |A|  =  0.

Hence it is singular matrix.

Question 2 :

Determine the values of a and b so that the following matrices are singular:

Since it is singular matrix, |A|  =  0

|A|  =  7a - (-6)  =  0

  7a + 6  =  0

  7a  =  -6

  a  =  -6/7

Hence the value of a is -6/7.

Solution :

Since it is singular matrix, |B|  =  0

|B|  =  (b- 1)[4 + 4] - 2[12 - 2] + 3[-6 - 1]

(b - 1)(8) - 2(10) + 3(-7)  =  0

8(b - 1) - 20 - 21  =  0

8(b - 1) - 41  =  0

8(b-1)  =  41

b-1  =  41/8

b  =  (41/8) + 1

   =  (41 + 8)/8  =  49/8

Hence the value of B is 49/8.

Question 3 :

If cos 2 θ = 0 , determine

Solution :

cos 2 θ = 0 

2θ = cos^-1(0 )

2θ = 90 degree

θ  =  90/2  =  45 degree

To multiply the above determinants, let us use row by column rule.

  =  1(1 - sin²θcos²θ) - sinθcosθ(sinθcosθ-sin²θcos²θ) + sinθcosθ(sin²θcos²θ-sinθcosθ)

  =  1 - sin²θcos²θ - sin²θcos²θ + sin³θcos³θ + sin³θcos³θ - sin²θcos²θ 

  =  1 - 3sin²θcos²θ + 2sin³θcos³θ

  =  1 - 3(sinθcosθ)² + 2(sinθcosθ)³

By applying 45 degree instead of θ, we get

  =  1 - 3(sin 45 cos 45)² + 2(sin 45 cos 45)³

  =  1 - 3((1/√2)(1/√2))² + 2((1/√2)(1/√2))³

  =  1 - 3(1/4) + 2(1/8)

  =  1 - (3/4) + (1/4)

  =  (4 - 3 + 1)/4

  =  2/4  =  1/2

Hence the answer is 1/2.

Question 4 :

Find the value of product 

Solution :

In order to find the square of the given determinant, we have to multiply the given determinant by the same.

Here we have followed row by column multiplication.

  =  21 - 15

  =  6

Hence the answer is 6.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.