Inverse Matrix and Invertible Matrix

If A is a square matrix of order m , and if there exists another square matrix B of the

same order m , such that AB = BA = I ,

then B is called the inverse matrix of A and it is denoted by $A^{-1}$ . In that case A is
said to be invertible matrix.

e.g. : Let

$A=\begin{bmatrix} 2 & 3\\ 1 & 2 \end{bmatrix} \ and \ B=\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix} \ be \ two \ matrices$

$Now, \ \ \ AB=\begin{bmatrix} 2 & 3\\ 1 &2 \end{bmatrix}\begin{bmatrix} 2 & -3\\ -1 &2 \end{bmatrix}$

$=\begin{bmatrix} 4-3 & -6+6\\ 2-2&-3+4 \end{bmatrix}=\begin{bmatrix} 1 & 0\\ 0 &0 \end{bmatrix}=I$

$Also \ \ \ \ BA=\begin{bmatrix} 1 & 0\\ 0 &0 \end{bmatrix}=I$
Thus B is the inverse of A , in other words $B=A^{-1}$ and A is inverse of B
i.e., $A=B^{-1}$

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Inverse Matrix and Invertible Matrix

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