**How to Identify If the Given Matrix is Singular or Nonsingular :**

Here we are going to see, how to check if the given matrix is singular or non singular.

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.

= 1[45-48]-2[36-42]+3[32-35] = 1[-3] - 2[-6] + 3[-3] = -3 + 12 - 9 = 0 |

Hence the matrix is singular matrix.

**Solution :**

= 2[0-20]+3[42-4]+5[30-0] = 2(-20) + 3(38) + 5(30) = -40 + 84 + 150 = 194 |

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 - sin^{2}θcos^{2}θ) - sinθcosθ(sinθcosθ-sin^{2}θcos^{2}θ) + sinθcosθ(sin^{2}θcos^{2}θ-sinθcosθ)

= 1 - sin^{2}θcos^{2}θ - sin^{2}θcos^{2}θ + sin^{3}θcos^{3}θ + sin^{3}θcos^{3}θ - sin^{2}θcos^{2}θ

= 1 - 3sin^{2}θcos^{2}θ + 2sin^{3}θcos^{3}θ

= 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.

After having gone through the stuff given above, we hope that the students would have understood, "How to Identify If the Given Matrix is Singular or Nonsingular".

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