Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}1&0\\0&1\end{bmatrix},\text{B}\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}0&1\\1&0\end{bmatrix},$ then show that $A^2 = B^2 = C^2 = l_2$​​​​​​​.
✓

Answer

Here,
$\text{A}^2=\text{AA}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1+0&0+0\\0+0&0+1\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\ \dots(1)$
$\text{B}^2=\text{BB}$
$\Rightarrow\text{B}^2=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}$
$\Rightarrow\text{B}^2=\begin{bmatrix}1+0&0-0\\0-0&0+1\end{bmatrix}$
$\Rightarrow\text{B}^2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\ \dots(2)$
$\text{C}^2=\text{CC}$
$\Rightarrow\text{C}^2=\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}$
$\Rightarrow\text{C}^2=\begin{bmatrix}0+1&0+0\\0+0&1+0\end{bmatrix}$
$\Rightarrow\text{C}^2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\ \dots(3)$
We know,
$\text{I}_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\ \dots(4)$
$\Rightarrow\text{A}^2=\text{B}^2=\text{C}^2=\text{I}^2$ [From eqs. (1), (2), (3) and (4)]

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

Find the shortest distance between the lines $\frac{\text{x}-2}{-1}=\frac{\text{y}-5}{2}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}-0}{2}=\frac{\text{y}+5}{-1}=\frac{\text{z}-1}{2}.$
→
If $(\text{x}-\text{y})\text{e}^{\frac{\text{x}}{\text{x}-\text{y}}}=\text{a},$ prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=2\text{y}$
→
Differentiate $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big)$ with respect to $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$
→
If $\hat{\text{a}}$ and $\hat{\text{b}}$ are unit vectors inclined at an angle $\theta$, prove that$\tan\frac{\theta}{2}=\frac{\big|\hat{\text{a}}-\hat{\text{b}}\big|}{\big|\hat{\text{a}}+\hat{\text{b}}\big|}$
→
Solve the following systems of linear equations by cramer's rule:
5x - 7y + z = 11,
6x - 8y - z = 15,
3x + 2y - 6z = 7
→
Evaluate the following integrals:
$\int\limits^{1}_0\big(\cos^{-1}\text{x}\big)^2\text{dx}$
→
Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{ x, y}(0)=1$
→
An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting,
  1. 2 red balls,
  2. 2 blue balls,
  3. One red and one blue ball.
→
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$x^5 + y^5 = 5xy$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2}{1-\text{x}^4}\ \text{dx}$
→