Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}0&-1\\1&0\end{bmatrix},$ then show that $(\text{A}+\text{B})(\text{A}-\text{B})\neq\text{A}^2-\text{B}^2.$
✓

Answer

We have, $\text{A}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$\therefore\ (\text{A}+\text{B})=\begin{bmatrix}0+0&1-1\\1+1&1+0\end{bmatrix}$
$=\begin{bmatrix}0&0\\2&1\end{bmatrix}$
and $(\text{A}-\text{B})=\begin{bmatrix}0-0&1+1\\1-1&1-0\end{bmatrix}$
$=\begin{bmatrix}0&2\\0&1\end{bmatrix}$
$(\text{A}+\text{B}).(\text{A}-\text{B})=\begin{bmatrix}0&0\\2&1\end{bmatrix}\begin{bmatrix}0&2\\0&1\end{bmatrix}$
$=\begin{bmatrix}0+0&0+0\\0+0&4+1\end{bmatrix}=\begin{bmatrix}0&0\\0&5\end{bmatrix}\ ....(\text{i})$
Also, $\text{A}^2=\text{A}.\text{A}$
$=\begin{bmatrix}0&1\\1&1\end{bmatrix}.\begin{bmatrix}0&1\\1&1\end{bmatrix}$
$=\begin{bmatrix}0+1&0+1\\0+1&1+1\end{bmatrix}=\begin{bmatrix}1&1\\1&2\end{bmatrix}$
and $\text{B}^2=\text{B}.\text{B}$
$=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}0&-1\\1&0\end{bmatrix}$
$=\begin{bmatrix}0-1&0+0\\0+0&-1+0\end{bmatrix}=\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$
$\therefore\ \text{A}^2-\text{B}^2=\begin{bmatrix}1&1\\1&2\end{bmatrix}-\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$
$=\begin{bmatrix}2&1\\1&3\end{bmatrix}\ ....(\text{ii})$
From (i) and (ii), $(\text{A}+\text{B})(\text{A}-\text{B})\neq\text{A}^2-\text{B}^2$

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 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

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\sqrt{\cos\text{x}-\cos^3\text{x}}(\sec^3\text{x}-1)\cos^2\text{x dx}$
→
If the area bounded by the parabola $\text{y}^{2} = 16\text{ax}$ and the line $\text{y = 4 mx}$ is $\frac{\text{a}^{2}}{12}$ sq. units, then using integration, find the value of m.
→
If the points $(x, -2), (5, 2), (8, 8)$ are collinear, find $x$ using determinants.
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{\text{x}(\text{x}^2-1)}\ \text{dx}$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}|\text{x}-3|,&\text{if }\text{ x}\geq1\\\frac{\text{x}^2}{4}-\frac{3\text{x}}{2}+\frac{13}{4},&\text{if }\text{ x}<1\end{cases}$
→
Prove the following identities:
$\begin{vmatrix}2\text{y}&\text{y}-\text{z}-\text{x}&2\text{y}\\2\text{z}&2\text{z}&\text{z}-\text{x}-\text{y}\\\text{x}-\text{y}-\text{z}&2\text{x}&2\text{x}\end{vmatrix}$
$=(\text{x}+\text{y}+\text{z})^3$
→
Solve the following differential equation:
$(\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=2\sin^2\text{x}\cos\text{x}$
→
A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.
→
Let $X $ denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of $X.$
→
Evaluate the following integrals:
$\int\limits^{\text{a}}_0\frac{1}{\text{x}+\sqrt{\text{a}^2-\text{x}^2}}\text{ dx}$
→