Download App

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

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAdjoint and Inverse of a Matrix4 Marks
Question
Find the inverse of the following matrices by using elementry row transformation:

$\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$

✓

Answer

$\text{A}=\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$

We have A = IA

$\Rightarrow\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$

$\Rightarrow\begin{bmatrix} 3 & -3 & \frac{4}{3} \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}=\begin{bmatrix} \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$

$\Big[\text{Applying R}_1\rightarrow\frac{1}{3}\text{R}_1\Big]$

$\Rightarrow\begin{bmatrix} 1 & -1 & \frac{4}{3} \\ 0 & -1 & \frac{4}{3} \\ 0 & -1 & 1 \end{bmatrix}=\begin{bmatrix} \frac{1}{3} & 0 & 0 \\ \frac{-2}{3} & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$

$\big[\text{Applying R}_2\rightarrow\text{R}_2-2\text{R}_1\big]$

$\Rightarrow\begin{bmatrix} 1 & -1 & \frac{4}{3} \\ 0 & 1 & \frac{-4}{3} \\ 0 & -1 & 1 \end{bmatrix}=\begin{bmatrix} \frac{1}{3} & 0 & 0 \\ \frac{2}{3} & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$

$\big[\text{Applting R}_2\rightarrow-\text{R}_2\Big]$

$\Rightarrow\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac{-4}{3} \\ 0 & 0 & \frac{-1}{3} \end{bmatrix}=\begin{bmatrix} 1 & -1 & 0 \\ \frac{2}{3} & -1 & 0 \\ \frac{2}{3} & -1 & 1 \end{bmatrix}\text{A}$

$\big[\text{Applying R}_1\rightarrow\text{R}_1+\text{R}_2\text{ and R}_3\rightarrow\text{R}_3+\text{R}_3\big]$

$\Rightarrow\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac{-4}{3} \\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & -1 & 0 \\ \frac{2}{3} & -1 & 0 \\ -2 & 3 & -3 \end{bmatrix}\text{A}$

$\big[\text{Applying R}_3\rightarrow-3\text{R}_3\big]$

$\Rightarrow\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & -1 & 0 \\ -2 & -1 & -4 \\ -2 & 3 & -3 \end{bmatrix}\text{A}$

$\big[\text{Applying R}_2\rightarrow\text{R}_2+\frac{4}{3}\text{R}_3\big]$

$\therefore\ \text{A}^{-1}=\begin{bmatrix} 1 & -1 & 0 \\ -2 & 3 & -4 \\ -2 & 3 & -3 \end{bmatrix}$

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 "4 Marks" from Adjoint and Inverse of a Matrix→Adjoint and Inverse of a Matrix — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equations:

$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{ x, y}(0)=1$

→
Show that the lines $\vec{\text{r}}=(2\hat{\text{i}}-3\hat{\text{k}})+\lambda(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$ and $\vec{\text{r}}=(2\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}})+\mu(2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})$ are coplanar. Also, find the equation of the plane containing them.
→
Evaluate the following integrals:

$\int\frac{1}{\sqrt{(\text{x}-\alpha)(\beta-\text{x})}}\text{ dx},(\beta>\alpha)$

→
Show that the matrix $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ satisfies the equation A3 - 4A2 + A = 0.
→
Evaluate the following integrals as limit of sum:
$\int\limits^{3}_{2}\text{x}^2\text{ dx}$
→
Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$
→
Evaluate :
$
\int_0^{\frac{\pi}{4}}\left(\frac{\sin x+\cos x}{3+\sin 2 x}\right) d x
$
→
If $=\text{y}=(\sin^{-1}\text{x})^2,$ prove that $(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-2=0.$
→
Find the area, lying above x-axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.
→
Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 : 4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C.
→