Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES4 Marks
Question
Using elementary transformations, find the inverse of the matrix
$\begin{pmatrix} 1 & 3 & -2 \\ -3 & 0 & -1\\ 2 & 1 & 0 \end{pmatrix}.$
✓

Answer

Given matrix A can be written as
$\begin{pmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\text{A}$
$\text{Applying R}_{2}\rightarrow\text{R}_{2}+\text{3R}_{1}\text{ R}_{3}\rightarrow{\text{R}_{3}-\text{2R}_{1}}$ $\begin{pmatrix} 1 & 3 & -2 \\ 0 & 9 & -7 \\ 0 & -5 & 4 \end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ -2 & 0 & 1 \end{pmatrix}\text{A}$
$\text{R}_{2}\rightarrow\text{R}_{2}+\text{2R}_{3}\Rightarrow\begin{pmatrix} 1 & 3 & -2 \\ 0 & -1& 1 \\ 0 & -5 & 4 \end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 2 \\ -2 & 0 & 1 \end{pmatrix}\text{A}$
$\text{R}_{3}\rightarrow\text{R}_{3}-\text{5R}_{2}\Rightarrow\begin{pmatrix} 1 & 3 & -2 \\ 0 & -1& 1 \\ 0 & 0 & -1 \end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 2 \\ 3 & -5 & -9 \end{pmatrix}\text{A}$
$\text{R}_{2}\rightarrow\text{R}_{2}+\text{R}_{3},\text{R}_{3}\rightarrow\begin{pmatrix} 1 & 3 & -2 \\ 0 & -1& 0 \\ 0 & 0 & 1 \end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ 2 & -4 & -7 \\ -3 & 5 & 9 \end{pmatrix}\text{A}$
$\text{R}_{1}\rightarrow\text{R}_{1}+\text{2R}_{3},\text{R}_{2}\rightarrow\text{-R}_{3}\begin{pmatrix} 1 & 3 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{pmatrix}=\begin{pmatrix} -5 & 10 & 18 \\ -2 & 4 & 7 \\ -3 & 5 & 9 \end{pmatrix}\text{A}$
$\text{R}_{1}\rightarrow\text{R}_{1}-\text{3R}_{2}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{pmatrix}=\begin{pmatrix} 1 & -2 & -3 \\ -2 & 4 & 7 \\ -3 & 5 & 9 \end{pmatrix}\text{A}$
$\therefore\text{A}^{-1}\begin{pmatrix} 1 & -2 & -3 \\ -2 & 4& 7 \\ -3 & 5 & 9 \end{pmatrix}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from MATRICES→MATRICES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3).
→
Evaluate the following integrals:

$\int\frac{\text{ax}^3+\text{bx}}{\text{x}^4+\text{c}^2}\text{ dx}$

→
A man rides his motorcycle at the speed of 50km/ hour. He has to spend Rs. 2 per km on petrol. If he rides it at a faster speed of 80km/ hour, the petrol cost increases to Rs. 3 per km. He has atmost Rs. 120 to spend on petrol and one hour’s time. He wishes to find the maximum distance that he can travel.
Express this problem as a linear programming problem.
→
Show that $\text{y}=\text{e}^\text{x}(\text{A}\cos\text{x}+\text{B}\sin\text{x})$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$
→
Solve the following differential equations:

$\frac{\text{dy}}{\text{dx}}=2\text{e}^{2\text{x}}\text{y}^2,\text{y}(0)=-1$

→
Solve the following equation for x:
$\tan^{-1}\frac{\text{x}-2}{\text{x}-1}+\tan^{-1}\frac{\text{x}+2}{\text{x}+1}=\frac{\pi}{4}$
→
Find the distance between the point (-1, -5, -10) and the point of intersection of the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}$ and the plane $x - y + z = 5.$
→
Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(3\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\big)$ and $\vec{\text{r}}=-\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(7\hat{\text{i}}-6\hat{\text{j}}+\hat{\text{k}}\big)$
→
Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
→
If $\text{a}+\text{b}+\text{c}\neq0\text{ and}\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{b}&\text{c}&\text{a}\\\text{c}&\text{a}&\text{b}\end{vmatrix}=0,$ then prove that a = b = c.
→