Inverse matrix

Inverse matrix

The transpose of the matrix obtained by replacing the element of the square matrix A by the corresponding cofactor is called adjoint of the matrix of A and it denote by adj(A)

A square matrix A is said to be an invertible matrix if there exist a square matrix B such that AB = BA = I the matrix B is called invers of A matrix and it denoted by A⁻¹.

If A is an inversible then  \({{A}^{-1}}=\frac{adj(A)}{\det (A)}\) .

Example: find the adjoint and inverses of the following matrices

1) \(A=\left[ \begin{matrix}   2 & -3  \\   4 & 6  \\\end{matrix} \right]\).

Solution: Given that \(A=\left[ \begin{matrix}   2 & -3  \\   4 & 6  \\\end{matrix} \right]\),

\(A=\left[ \begin{matrix}   a & b  \\   c & d  \\\end{matrix} \right]=\left[ \begin{matrix}   2 & -3  \\   4 & 6  \\\end{matrix} \right]\),

\(adj(A)=\left[ \begin{matrix}   d & -b  \\   -c & d  \\\end{matrix} \right]=\left[ \begin{matrix}6 & 3  \\   -4 & 2  \\\end{matrix} \right]\),

|A| = 12 – (- 12) = 24

\({{A}^{-1}}=\frac{adj(A)}{\det (A)}=\frac{1}{24}\left[ \begin{matrix} 6 & 3  \\   -4 & 2  \\\end{matrix} \right]\).

2) \(A=\left[ \begin{matrix}   1 & 0  \\   0 & 1  \\\end{matrix} \right]\).

Solution: Given that \(A=\left[ \begin{matrix}   1 & 0  \\   0 & 1  \\\end{matrix} \right]\),

\(A=\left[ \begin{matrix}   a & b  \\   c & d  \\\end{matrix} \right]=\left[ \begin{matrix}   1 & 0  \\   0 & 1  \\\end{matrix} \right]\),

\(adj(A)=\left[ \begin{matrix}   d & -b  \\   -c & d  \\\end{matrix} \right]=\left[ \begin{matrix}   1 & 0  \\   0 & 1  \\\end{matrix} \right]\),

|A| = 1 – 0 = 1

\({{A}^{-1}}=\frac{adj(A)}{\det (A)}=\frac{1}{1}\left[ \begin{matrix}   1 & 0  \\   0 & 1  \\\end{matrix} \right]\),

\({{A}^{-1}}=\left[ \begin{matrix}   1 & 0  \\   0 & 1  \\\end{matrix} \right]\).