Question

Find the inverse of the matrix \(\begin{bmatrix} 1& -3 \\ 4& 2 \end{bmatrix}\)
1. \({1\over14}\begin{bmatrix} 2& -3 \\ 4& -1 \end{bmatrix}\)
2. \({1\over14}\begin{bmatrix} 2& 3 \\ -4& 1 \end{bmatrix}\)
3. \({1\over14}\begin{bmatrix} 2& -3 \\ -4& 1 \end{bmatrix}\)
4. \({1\over14}\begin{bmatrix} 2& 3 \\ 4& 1 \end{bmatrix}\)
5. None of these

Answer 1

Correct Answer - Option 2 : \({1\over14}\begin{bmatrix} 2& 3 \\ -4& 1 \end{bmatrix}\)

Concept: 

For any matrix A, If matrix A = [aij]

A-1 = \(\rm \text{(adj A)}\over|A|\)

The adj A matrix is the transpose of the matrix [Aij] is the cofactor of the element aij.

Calculation:

A = \(\begin{bmatrix} 1& -3 \\ 4& 2 \end{bmatrix}\) = [aij]

|A| = \(\begin{vmatrix} 1& -3 \\ 4& 2 \end{vmatrix}\) 

⇒ |A| = 2 × 1 - (-3) × 4

⇒ |A| = 14

Now, finding cofactor of matrix

a₁₁ = 1, cof(a11) = 2

a₁₂ = -3, cof(a12) = -4

a₂₁ = 4, cof(a21) = 3

a₂₂ = 2, cof(a22) = 1

Cof(A) = \(\begin{bmatrix} 2& -4 \\ 3& 1 \end{bmatrix}\)

adj(A) = [cof(A)]^T 

adj(A) = \(\begin{bmatrix} 2& 3 \\ -4& 1 \end{bmatrix}\)

A^-1 = \(\rm \text{(adj A)}\over|A|\) 

⇒ A^-1 = \(\boldsymbol{{1\over14}\begin{bmatrix} 2& 3 \\ -4& 1 \end{bmatrix}}\)