Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Obtain the inverse of the following matrix using elementary operations: $\text{A}=\begin{bmatrix}-1 & 1 & 2\\1 & 2 & 3\\3 & 1 & 1 \end{bmatrix}$
✓

Answer

$\therefore\text{A}^{-1}=\text{IA}$
$\text{A}^{-1}=\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}\begin{bmatrix}-1 & 1 & 2\\1 & 2 & 3\\3 & 1 & 1 \end{bmatrix}$
$\text{R}_1\rightarrow-\text{R}_1$
$\text{A}^{-1}=\begin{bmatrix}-1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}\begin{bmatrix}1 & -1 & -2\\1 & 2 & 3\\3 & 1 & 1 \end{bmatrix}$
$\text{R}_2\rightarrow\text{R}_2-\text{R}_1$
$\text{R}_3\rightarrow\text{R}_3-3\text{R}_1$
$\text{A}^{-1}=\begin{bmatrix}-1 & 0 & 0\\1 & 1 & 0\\3 & 0 & 1 \end{bmatrix}\begin{bmatrix}1 & -1 & -2\\0 & 3 & 5\\0 & 4 & 7 \end{bmatrix}$
$\text{R}_2\rightarrow-\text{R}_2+\text{R}_3$
$\text{A}^{-1}=\begin{bmatrix}-1 & 0 & 0\\2 & -1 & 1\\3 & 0 & 1 \end{bmatrix}\begin{bmatrix}1 & -1 & -2\\0 & 1 & 2\\0 & 4 & 7 \end{bmatrix}$
$\text{R}_1\rightarrow\text{R}_1+\text{R}_2$
$\text{R}_3\rightarrow\text{R}_3-4\text{R}_2$
$\text{A}^{-1}=\begin{bmatrix}1 & 0 & 0\\0 & 1 & 2\\0 & 0 & -1 \end{bmatrix}\begin{bmatrix}1 & -1 & 1\\2 & -1 & 1\\-5 & 4 & -3 \end{bmatrix}$
$\text{R}_3\rightarrow-\text{R}_3$
$\text{A}^{-1}=\begin{bmatrix}1 & -1 & 1\\2 & -1 & 1\\5 & -4 & 3 \end{bmatrix}\begin{bmatrix}1 & 0 & 0\\0 & 1 & 2\\0 & 0 & 1 \end{bmatrix}$
$\text{R}_2\rightarrow\text{R}_2-2\text{R}_3$
$\text{A}^{-1}=\begin{bmatrix}1 & -1 & 1\\-8 & 7 & -5\\5 & -4 & 3 \end{bmatrix}\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}$
Here $\text{I}=\begin{bmatrix}1 & -1 & 1\\-8 & 7 & -5\\5 & -4 & 3 \end{bmatrix}$
So, $\text{A}^{-1}==\begin{bmatrix}1 & -1 & 1\\-8 & 7 & -5\\5 & -4 & 3 \end{bmatrix}$

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 "5 Marks Questions" from DETERMINANTS→DETERMINANTS — 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 integrals of the functions in Exercises:
$\sin\text{x } \sin2\text{x }\sin3\text{x}$
→
Find the particular solution of the differential equation dy = cos x (2 – y cosec x) dx, given that y = 2 when$\text{x} = \frac{\pi}{2}.$
→
Show that the lines $\frac{5-\text{x}}{-4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{2\text{y}-8}{2}=\frac{\text{z}-5}{3}$ are coplanar.
→
Evaluate the following:
$\int\frac{\sin^6\text{x}+\cos^6\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}$
→
Solve the following differential equation
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
→
Evaluate the following integrals:
$\int\limits^{\text{a}}_0\frac{1}{\text{x}+\sqrt{\text{a}^2-\text{x}^2}}\text{ dx}$
→
A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him ₹ 360 and a manually operated sewing machine ₹ 240. He can sell the sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as an LPP and solve it graphically.
→
A circular metal plate expends under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10cm.
→
Prove that:
$\begin{vmatrix}\frac{\text{a}^2+\text{b}^2}{\text{c}}&\text{c}&\text{c}\\\text{a}&\frac{\text{b}^2+\text{c}^2}{\text{a}}&\text{a}\\\text{b}&\text{b}&\frac{\text{c}^2+\text{a}^2}{\text{b}}\end{vmatrix}=4\text{abc}$
→
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
→