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Adjoint and Inverse of a Matrix — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAdjoint and Inverse of a Matrix5 Marks
Question
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$
✓

Answer

$\text{A}=\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$We know A = IA
$\Rightarrow\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$
$\Rightarrow\begin{bmatrix} 1 & 3 & -2 \\ 0 & 9 & -7 \\ 0 & -5 & 4 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ -2 & 0 & 1 \end{bmatrix}\text{A}$
$\big[\text{Applying R}_2\rightarrow\text{R}_2+3\text{R}_1\text{ and R}_3\rightarrow\text{R}_3-2\text{R}_1\big]$
$\begin{bmatrix} 1 & 3 & -2 \\ 0 & 1 & \frac{-7}{9} \\ 0 & -5 & 4 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ \frac{1}{3} & \frac{1}{9} & 0 \\ -2 & 0 & 1 \end{bmatrix}\text{A}$
$\Big[\text{Applying R}_2\rightarrow\frac{1}{2}\text{R}_2\Big]$
$\begin{bmatrix} 1 & 3 & \frac{1}{3} \\ 0 & 1 & \frac{-7}{9} \\ 0 & 0 & \frac{1}{9} \end{bmatrix}=\begin{bmatrix} 0 & -\frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{9} & 0 \\ -\frac{1}{3} & \frac{5}{9} & 1 \end{bmatrix}\text{A}$
$\big[\text{Applying R}_1\rightarrow\text{R}_1-3\text{R}_2\text{ and R}_3\rightarrow\text{R}_3\rightarrow\text{R}_3+5\text{R}_2\big]$
$\Rightarrow\begin{bmatrix} 1 & 0 & \frac{1}{3} \\ 0 & 1 & \frac{-7}{9} \\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 0 & -\frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{9} & 0 \\ -3 & 5 & 9 \end{bmatrix}\text{A}$
$\big[\text{Applying R}_3\rightarrow9\text{R}_3\big]$
$\Rightarrow\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & -2 & -3 \\ -2 & 4 & 7 \\ -3 & 5 & 9 \end{bmatrix}\text{A}$
$\Big[\text{Applying R}_2\rightarrow\text{R}_2+\frac{7}{9}\text{R}_3\text{ and R}_1\rightarrow\text{R}_1-\frac{1}{3}\text{R}_3\Big]$
$\Rightarrow\text{A}^{-1}=\begin{bmatrix} 1 & -2 & -3 \\ -2 & 4 & 7 \\ -3 & 5 & 9 \end{bmatrix}$

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