M.4 Matrix Inverse

Inverse of a Matrix

The matrix B is the inverse of matrix A if $A B = B A = I$ . This is often denoted as $B = A^{- 1}$ or $A = B^{- 1}$ . When taking the inverse of the product of two matrices A and B,

$(A B)^{- 1} = B^{- 1} A^{- 1}$

When taking the determinate of the inverse of the matrix A,

$d e t (A^{- 1}) = \frac{1}{d e t (A)} = d e t (A)^{- 1}$

Note that not all matrices have inverses. For a matrix A to have an inverse, that is to say for A to be invertible, A must be a square matrix and $d e t (A) \neq 0$ . For that reason, invertible matrices are also called nonsingular matrices.

Two examples are shown below

$d e t (A) = | \begin{matrix} 4 & 5 \\ - 2 & 1 \end{matrix} | = 4 * 1 - 5 * - 2 = 14 \neq 0$

$d e t (C) = | \begin{matrix} 1 & 2 & - 1 \\ 5 & 3 & 2 \\ 6 & 0 & 6 \end{matrix} | = - 2 | \begin{matrix} 5 & 2 \\ 6 & 6 \end{matrix} | + 3 | \begin{matrix} 1 & - 1 \\ 6 & 6 \end{matrix} | + 0 | \begin{matrix} 1 & - 1 \\ 5 & 2 \end{matrix} |$

$d e t (C) = - 2 (5 * 6 - 2 * 6) + 3 (1 * 6 - (- 1) * 6) - 0 (1 * 2 - (- 1) * 5) = 0$

So C is not invertible, because its determinate is zero. However, A is an invertible matrix, because its determinate is nonzero. To calculate that matrix inverse of a 2 × 2 matrix, use the below formula.

$A^{- 1} = {(\begin{matrix} a_{1, 1} & a_{1, 2} \\ a_{2, 1} & a_{2, 2} \end{matrix})}^{- 1} = \frac{1}{d e t (A)} (\begin{matrix} a_{2, 2} & - a_{1, 2} \\ - a_{2, 1} & a_{1, 1} \end{matrix}) = \frac{1}{a_{1, 1} * a_{2, 2} - a_{1, 2} * a_{2, 1}} (\begin{matrix} a_{2, 2} & - a_{1, 2} \\ - a_{2, 1} & a_{1, 1} \end{matrix})$

For example

$A^{- 1} = {(\begin{matrix} 4 & 5 \\ - 2 & 1 \end{matrix})}^{- 1} = \frac{1}{d e t (A)} (\begin{matrix} 1 & - 5 \\ 2 & 4 \end{matrix}) = \frac{1}{4 * 1 - 5 * (- 2)} (\begin{matrix} 1 & - 5 \\ 2 & 4 \end{matrix}) = (\begin{matrix} \frac{1}{14} & \frac{- 5}{14} \\ \frac{2}{14} & \frac{4}{14} \end{matrix})$

For finding the matrix inverse in general, you can use Gauss-Jordan Algorithm. However, this is a rather complicated algorithm, so usually one relies upon the computer or calculator to find the matrix inverse.

^[1]	Link
↥	Has Tooltip/Popover
	Toggleable Visibility