Question

If \( A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 2 & -3 \\ 1 & 1 & 1\end{array}\right], f i n d(\operatorname{adi} . A) \) Check whether \( A \) is singular or not.

Answer 1

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

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

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

 = 5 + 5 = 10 ≠ 0

\(\therefore\) Matrix A is not singular.

Cofactor of a₁₁ is C₁₁ = (-1)¹⁺¹M₁₁ = (-1)¹⁺¹\(\begin{vmatrix}2&-3\\1&1\end{vmatrix}\) = 2 + 3 = 5

Cofactor of a₁₂ is C₁₂ = (-1)¹⁺²\(\begin{vmatrix}2&-3\\1&1\end{vmatrix}\) = -(2 + 3) = -5

Cofactor of a₁₃ is C₁₃ = \(\begin{vmatrix}2&2\\1&1\end{vmatrix}\) = 2- 2 = 0

Cofactor of a₂₁ is C₂₁ = \(-\begin{vmatrix}-1&1\\1&1\end{vmatrix}\) = -(-1- 1) = 2

Cofactor of a₂₂ is C₂₂ = \(\begin{vmatrix}1&1\\1&1\end{vmatrix}\) = 1 - 1 = 0

Cofactor of a₂₃ is C₂₃ = \(-\begin{vmatrix}1&1\\1&1\end{vmatrix}\) = -(1 + 1) = -2

Cofactor of a₃₁ is C₃₁ = \(\begin{vmatrix}-1&1\\2&-3\end{vmatrix}\) = 3 - 2 = 1

Cofactor of a₃₂ is C₃₂ = \(\begin{vmatrix}1&1\\2&-3\end{vmatrix}\) = -(-3 - 2) = 5

Cofactor of a₃₃ is C₃₃ = \(\begin{vmatrix}1&-1\\2&2\end{vmatrix}\) = 2 + 2 = 4

\(\therefore\) Cofactor matrix of A is C = \(\begin{bmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\c_{31}&c_{32}&c_{33}\end{bmatrix}\)  = \(\begin{bmatrix}5&-5&0\\2&0&-2\\1&5&4\end{bmatrix}\) 

\(\therefore\) Adjoint of matrix A is adj A =C^T = \(\begin{bmatrix}5&2&1\\-5&5&-2\\0&-2&4\end{bmatrix}\)