Class 12th

Matrices A and B will be inverse of each other if.

(E) AB = BA = I.

Here option D is correct.

Let A = $\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 5\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 2-2(1)\hfill & \hfill 0-2(1)\hfill & \hfill -1-2(2)\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1-2(-2)\hfill & \hfill 0-2(1)\hfill & \hfill 0-2\left(0\right)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ A. (R2      R2  ----> 2R1)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill -5\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill -5\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 0\hfill \end{array}\right]$ A. (R2  <-->R3)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0+2(0)\hfill & \hfill -2+2(1)\hfill & \hfill -5+2(3)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 5+2(0)\hfill & \hfill -2+2(0)\hfill & \hfill 0+2(1)\hfill \end{array}\right]$ A. (R3 ->R3 + 2R2)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1-2(0)\hfill & \hfill 1-2(0)\hfill & \hfill 2-2(1)\hfill \\ \hfill 0-3(0)\hfill & \hfill 1-3(0)\hfill & \hfill 3-3(1)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -2-2(5)\hfill & \hfill 1-2(-2)\hfill & \hfill 0-2(2)\hfill \\ \hfill 0-3(5)\hfill & \hfill 0-3(-2)\hfill & \hfill 1-3(2)\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$ A. $\left(\begin{array}{ll}{R}_1\to R_1-2R_3\hfill & R_2\to R_2-3R_3\hfill \end{array}\right)$

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -12\hfill & \hfill 5\hfill & \hfill -4\hfill \\ \hfill -15\hfill & \hfill 6\hfill & \hfill -5\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1-0\hfill & \hfill 1-1\hfill & \hfill 0-0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -12-(+15)\hfill & \hfill 5-6\hfill & \hfill -4-(-5)\hfill \\ \hfill -15\hfill & \hfill 6\hfill & \hfill -5\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$ A. (R1 --->R1 - R2).

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill -15\hfill & \hfill 6\hfill & \hfill -5\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$ A.

∴ A1 = $\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill -15\hfill & \hfill 6\hfill & \hfill -5\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$

Let A = $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill -3\hfill & \hfill 0\hfill & \hfill -5\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 0\hfill \end{array}\right]$

We write A = IA

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill -3\hfill & \hfill 0\hfill & \hfill -5\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill -3+3(1)\hfill & \hfill 0+3(3)\hfill & \hfill -5+3(-2)\hfill \\ \hfill 2-2(1)\hfill & \hfill 5-2(3)\hfill & \hfill 0-2(-2)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0+3(1)\hfill & \hfill 1+0\hfill & \hfill 0+0\hfill \\ \hfill 0-2(1)\hfill & \hfill 0-0\hfill & \hfill 1-0\hfill \end{array}\right]$ A. $\left(\begin{array}{ll}{R}_2\to R_2+3R_1\hfill & R_3\to R_3-2R_1\hfill \end{array}\right)$

$=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 9\hfill & \hfill -11\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill .4\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ A.

$\to \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 4p\hfill \\ \hfill 0\hfill & \hfill 9\hfill & \hfill -11\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ A. (R2        R3)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill 0\hfill & \hfill 9\hfill & \hfill -11\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ A. (R2   (-1) R2)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill 0-0\hfill & \hfill 9-9(1)\hfill & \hfill -11-9(-4)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 3-9(2)\hfill & \hfill 1-0\hfill & \hfill 0-9(-1)\hfill \end{array}\right]$ A. (R3     R3--> 4R2)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 25\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill -15\hfill & \hfill 1\hfill & \hfill 9\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A. $\left(R_3\to \frac{1}{25}{R}_3\right)$

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0+0\hfill & \hfill 1+0\hfill & \hfill -4+4(1)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=$ $=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill \hfill \\ \hfill 2+4\left(\frac{3}{5}\right)\hfill & \hfill 0+4\left(\frac{1}{25}\right)\hfill & \hfill -1+4*\left(\frac{9}{25}\right)\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A. (R2     R2 + 4R3)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1+0\hfill & \hfill 3+0\hfill & \hfill -2+2(1)\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1+2\left(-\frac{3}{5}\right)\hfill & \hfill 0+2\left(\frac{1}{25}\right)\hfill & \hfill 0+2\left(\frac{9}{23}\right)\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A. (R1  R1 + 2R3)

$\Rightarrow \left[\begin{array}{lll}1\hfill & 3\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$18$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1+0\hfill & \hfill 3-3(1)\hfill & \hfill 0+0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=$ $\left[\begin{array}{ccc}\hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.-3\left(\frac{-2}{5}\right)\hfill & \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$25$}\right.-3\left(\frac{4}{25}\right)\hfill & \hfill \frac{18}{25}-3\left(\frac{11}{25}\right)\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 25\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A. (R1   R1 3R2)

$\Rightarrow \left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \raisebox{1ex}{$-10$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$-15$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$ A.

∴A^-1 = $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \\ \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \\ \hfill \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\hfill \end{array}\right]$

Let A = $\left[\begin{array}{ccc}\hfill 2\hfill & \hfill -3\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]$

We write, A = IA

$\Rightarrow \left[\begin{array}{ccc}\hfill 2\hfill & \hfill -3\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill -2\hfill & \hfill 2\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 3\hfill & \hfill -2\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -3\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ A. $\left[R_3\leftrightarrow R_1\right)$

$\Rightarrow \left[\begin{array}{ccc}\hfill 3-2\hfill & \hfill -2-2\hfill & \hfill 2-3\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -3\hfill & \hfill 3\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill 0-0\hfill & \hfill 0-1\hfill & \hfill 1-0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ A. (R1   R1 -->R2)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -3\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 10\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ A.

$\Rightarrow \left|\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 2-2(1)\hfill & \hfill 2+2(-4)\hfill & \hfill 3-2(-1)\hfill \\ \hfill 2-2(1)\hfill & \hfill -3+2(-4)\hfill & \hfill 3-2(-1)\hfill \end{array}\right|=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 0-2(0)\hfill & \hfill 1-2(-1)\hfill & \hfill 0-2(1)\hfill \\ \hfill 1-2(0)\hfill & \hfill 0-2(-1)\hfill & \hfill 0-2(1)\hfill \end{array}\right]$ A.

$(\begin{array}{l}{R}_2\to R_2-2R_1\hfill \end{array}$

$\begin{array}{l}{R}_3\to R_3-2R_1\hfill \end{array})$

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 10\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 5\hfill & \hfill 5\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill -2\hfill \end{array}\right]$ A.

$\bcancel{1:\frac{x^2}{36}}+\bcancel{\frac{y^2}{16}=1}$

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 0-0\hfill & \hfill 10-5\hfill & \hfill 5-5\hfill \\ \hfill 0\hfill & \hfill 5\hfill & \hfill 5\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 0-1\hfill & \hfill 3-2\hfill & \hfill -2-(-2)\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill -2\hfill \end{array}\right]$ A. (R2   R2 --->R3)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 5\hfill & \hfill 5\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill -2\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A. $\left(\begin{array}{cc}\hfill R_2\to \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.R_2\hfill & \hfill R_3\to \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.R_3\hfill \end{array}\right)$

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0-0\hfill & \hfill 1-1\hfill & \hfill 1-0\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill 1/5-\left(\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)\hfill & \hfill \frac{2}{5}-\frac{1}{5}\hfill & \hfill 0\left(\raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)-0\hfill \end{array}\right]$ A. (R3       R3 -->R2)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1+0\hfill & \hfill -4+0\hfill & \hfill -1+1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill -1+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 1+\left(\raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A. (R1      R1 + R3)

$\Rightarrow \left[\begin{array}{ccc}\hfill 1\hfill & \hfill -4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A.

$\Rightarrow \left[\begin{array}{ccc}\hfill 1+0\hfill & \hfill -4+4(1)\hfill & \hfill 0+0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=$ $\left[\begin{array}{ccc}\hfill \frac{2}{5}+4\left(-\frac{1}{5}\right)\hfill & \hfill -\frac{4}{5}+4\left(\frac{1}{5}\right)\hfill & \hfill \frac{3}{5}+4(0)\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A. (R1      R1 + 4R2)

$\Rightarrow \left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill & \hfill \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$ A.

∴A^-1 = $\left[\begin{array}{ccc}\hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill & \hfill \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \\ \hfill \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill 0\hfill \\ \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \end{array}\right]$

Let A = $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 2\hfill \end{array}\right]$

We write, A = IA.

$\Rightarrow \left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ A.