Download App

Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
Question
If $\text{A}=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix},\text{B}=\begin{bmatrix}0&5&-4\\-2&1&3\\-1&0&2\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&5&2\\-1&1&0\\0&-1&1\end{bmatrix},$ verify that A(B - C) = AB - AC.
✓

Answer

Given, $\text{A}=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix},\text{B}=\begin{bmatrix}0&5&-4\\-2&1&3\\-1&0&2\end{bmatrix}$ $\text{C}=\begin{bmatrix}1&5&2\\-1&1&0\\0&-1&1\end{bmatrix}$ $ \text{A}(\text{B}-\text{C})=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix}\begin{bmatrix}\begin{bmatrix}0&5&-4\\-2&1&3\\-1&0&2\end{bmatrix}-\begin{bmatrix}1&5&2\\-1&1&0\\0&-1&1\end{bmatrix}\end{bmatrix}$ $=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix}\begin{bmatrix}0-1&5-5&-4-2\\-2+1&1-1&3-0\\-1-0&0+1&2-1\end{bmatrix}$ $=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix}\begin{bmatrix}-1&0&-6\\-1&0&3\\-1&1&1\end{bmatrix}$ $=\begin{bmatrix}-1+0+2&0+0-2&-6+0-2\\-3+1+0&0+0+0&-18-3+0\\2-1-1&0+0+1&12+3+1\end{bmatrix}$ $\text{A}(\text{B}-\text{C})=\begin{bmatrix}1&-2&-8\\-2&0&-21\\0&1&16\end{bmatrix}\ \dots(\text{i})$ $ \text{AB}-\text{AC}=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix}\begin{bmatrix}0&5&-4\\-2&1&3\\-1&0&2\end{bmatrix}-\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix}\begin{bmatrix}1&5&2\\-1&1&0\\0&-1&1\end{bmatrix}$ $=\begin{bmatrix}0+0+2&5+0+0&-4+0-4\\0+2+0&15-1+0&-12-3+0\\0-2-1&-10+1+0&8+3+2\end{bmatrix}-\begin{bmatrix}1+0+0&5+0+2&2+0-2\\3+1+0&15-1+0&6+0+0\\0-2-1&-10+1+1&-4+0+1\end{bmatrix}$ $=\begin{bmatrix}2&5&-8\\2&14&-15\\-3&-9&13\end{bmatrix}-\begin{bmatrix}1&7&0\\4&14&6\\-3&-10&-3\end{bmatrix}$ $=\begin{bmatrix}2-1&5-7&-8-0\\2-4&14-14&-14-6\\-3+3&-9+10&3+3\end{bmatrix}$ $\text{AB}-\text{AC}=\begin{bmatrix}1&-2&-8\\-2&0&-21\\0&1&16\end{bmatrix}\ \dots(\text{ii})$ From equation (i) and (ii),$\text{A}(\text{B}-\text{C})=\text{AB}-\text{AC}$

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 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

Discuss the applicability of Rolle’s theorem on the function given by.
$\text{f(x)}=\begin{cases}\text{x}^2+1,&\text{if }0\leq\text{x}\leq1\\3-\text{x},&\text{if }1\leq\text{x}\leq2\end{cases}$
→
Show that the lines $\frac{\text{x}-1}{3}=\frac{\text{y}+1}{2}=\frac{\text{z}-1}{5}$ and $\frac{\text{x}+2}{4}=\frac{\text{y}-1}{3}=\frac{\text{z}+1}{-2}$ do not intersect.
→
Solve the following system of equations by matrix method:
$x - y + 2z = 7$
$3x + 4y - 5z = -5$
$2x - y + 3z = 12$
→
If $\text{A}=\begin{bmatrix}\cos\alpha+\sin\alpha&\sqrt{2}\sin\alpha\\-\sqrt{2}\sin\alpha&\cos\alpha-\sin\alpha\end{bmatrix},$ prove that
$ \text{A}^2=\begin{bmatrix}\cos\text{n}\alpha+\sin\text{n}\alpha&\sqrt{2}\sin\text{n}\alpha\\-\sqrt{2}\sin\text{n}\alpha&\cos\text{n}\alpha-\sin\text{n}\alpha\end{bmatrix}$ for all $\text{n}\in\text{N}.$
→
If $x^x + y^x = 1$, prove that $\frac{\text{dy}}{\text{dx}}=-\Big\{\frac{\text{x}^\text{x}(1+\log\text{x})+\text{y}^\text{x}\times\log\text{y}}{\text{x}\times\text{y}^{\text{x}-1}}\Big\}$
→
$\text{Find}:\int\frac{(3\sin x-2)\cos x}{13\ -\ \cos^2x \ - \ 7\sin x}\text{d}x$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{ dx}$
→
Suppose a girl throw a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a 'head' or 'tail' is obtained exactly one 'tail', what is the probability that she threw 3, 4, 5 or 6 with the die?
→
Show that the function $\text{f(x)}=\begin{cases}|2\text{x}-3||\text{x}|, & \text{x}\geq1\\\sin\Big(\frac{\pi\text{x}}{2}\Big),& \text{x}>1\end{cases}$ is continuous but not differentiable at x = 1.
→
Prove that the area the first quadeant enclosed by the x-axis, the line $\text{x}=\sqrt{3}\text{y}$ and the circle is $\frac{\pi}{3}$ .
→