Download App

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:$\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$
✓

Answer

$\text{A}=\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$We know A = IA
$\Rightarrow\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\text{A}$
$\Rightarrow\begin{bmatrix} 1 & 0 & \frac{-1}{2} \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}=\begin{bmatrix}\frac{1}{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$
$\Big[\text{Applying R}_1\rightarrow\frac{1}{2}\text{R}_1\Big]$
$\Rightarrow\begin{bmatrix} 1 & 0 & \frac{-1}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 1 & 3 \end{bmatrix}=\begin{bmatrix} \frac{1}{2} & 0 & 0 \\ \frac{-5}{2} & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\text{A}$
$\big[\text{Applying R}_2\rightarrow\text{R}_2-5\text{R}_1\big]$
$\Rightarrow\begin{bmatrix} 1 & 0 & \frac{-1}{2} \\ 0 & 1 & \frac{5}{2}\\ 0 & 0 &\frac{1}{2} \end{bmatrix}=\begin{bmatrix}\frac{1}{2} & 0 & 0 \\ \frac{-5}{2} & 1 & 0 \\ \frac{5}{2} & -1 & 1\end{bmatrix}\text{A}$
$\big[\text{Applying R}_3\rightarrow\text{R}_3-\text{R}_2\big]$
$\Rightarrow\begin{bmatrix} 1 & 0 & \frac{-1}{2} \\ 0 & 1 & \frac{5}{2}\\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix}\frac{1}{2} & 0 & 0 \\ -\frac{5}{2} & 1 & 0 \\ 5 & -2 & 2\end{bmatrix}\text{A}$
$\big[\text{Applying R}_3\rightarrow\ 2\text{R}_3\big]$
$\Rightarrow\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}\text{A}$
$\Big[\text{Applying R}_1\rightarrow\text{R}_1+\frac{1}{2}\text{ R}_3\text{ and R}_2\rightarrow\text{R}_2-\frac{5}{2}\text{R}_3\Big]$
$\Rightarrow\text{A}^{-1}=\begin{bmatrix} 3 & -1 & 1\\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}$

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 "Solve the Following Question.(5 Marks)" from Adjoint and Inverse of a Matrix→Adjoint and Inverse of a Matrix — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{y}=(\cos\text{x})^{\cos\text{x}^{\cos\text{x}^{.....\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}^2\tan\text{x}}{(1-\text{y}\log\cos\text{x})}$
→
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=\text{a}^{\frac{2}{3}}$
→
Evaluate :

$\int \frac{7 x+3}{\sqrt{3+2 x-x^2}} d x$

→
If $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7\end{array}\right]$, then find a matrix $X$ such that $X A=B$.
→
Evaluate the following intregals:
$\int\frac{2\text{x}-3}{(\text{x}^2-1)(2\text{x}-3)}\ \text{dx}$
→
Show that the areas under the curves $\text{y}=\sin\text{x}\text{ and }\text{y}=\sin2\text{x}$ between x = 0 and $\text{x}=\frac{\pi}{3}$ are in the ratio 2 : 3.
→
Evaluate the following integral:
$\int\frac{\text{x}^4+1}{\text{x}^2+1}\text{ dx}$
→
Compute the adjoint of the following matrices:$\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$
Verify that (adjoint A)A = |A|I = A (adjoint A) for the above matrices.
→
Show that a closed right circular cylinder of a given surface area has maximum volume if its height equals the diameter of its base.
→
Find the equation of the plane mid-parallel to the planes $2x - 2y + z + 3 = 0$ and $2x - 2y + z + 9 = 0$
→