Download App

Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices4 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 A2 = B2 = C2 = l2.
✓

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 "4 Marks" from Algebra of Matrices→Algebra of Matrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\text{x}=\sin\text{t}$ and $\text{y}=\sin\text{pt},$ prove that $(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}+\text{p}^2\text{ y}=0.$
→
Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=1-\text{x + y}-\text{xy}$
→
If $\text{A}=\begin{bmatrix}1&2&0\\3&-4&5\\0&-1&3\end{bmatrix},$ compute A2 - 4A + 3I3.
→
A wholesale dealer deals in two kinds, A and B (say) of mixture of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew nuts and 540 grams of hazel nuts. Mixture A costs Rs. 8 per kg. and mixture B costs Rs. 12 per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg. of each mixture which he should use to minimise the cost of the bag.
→
Evaluate the following integrals:
$\int\limits_{0}^{\text{a}}\sqrt{\text{a}^2-\text{x}^2}\text{ dx}$
→
Using vector method, prove that the point is collinear:
A(6, -7, -1), B(2, -3, 1) and C(4, -5, 0)
→
Find the inverse of the matrix (if it exists) given $\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5\end{array}\right]$
→
Two tailors, A and B earn Rs. 15 and Rs. 20 per day respectively. A can stitch 6 shirts and 4 pants  while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost?
→
Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{\tan^{-1}\text{x}}{1+\text{x}^2}\text{ dx}$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\cot\text{x}}}{\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}}\text{ dx}$
→