Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}1&2\\-2&1\end{bmatrix},\ \text{B}=\begin{bmatrix}2&3\\3&-4\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&0\\-1&0\end{bmatrix},$ verify $(\text{AB})\text{C}=\text{A}(\text{BC}).$
✓

Answer

We have, $\text{A}=\begin{bmatrix}1&2\\-2&1\end{bmatrix},\ \text{B}=\begin{bmatrix}2&3\\3&-4\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&0\\-1&0\end{bmatrix}$
$\text{AB}=\begin{bmatrix}1&2\\-2&1\end{bmatrix}\begin{bmatrix}2&3\\3&-4\end{bmatrix}$
$=\begin{bmatrix}2+6&3-8\\-4+3&-6-4\end{bmatrix}=\begin{bmatrix}8&-5\\-1&-10\end{bmatrix}$
and $(\text{AB})\text{C}=\begin{bmatrix}8&-5\\-1&-10\end{bmatrix}\begin{bmatrix}1&0\\-1&0\end{bmatrix}$
$\begin{bmatrix}8+5&0\\-1+10&0\end{bmatrix}=\begin{bmatrix}13&0\\9&0\end{bmatrix}\ ....(\text{i})$
Again, $(\text{BC})=\begin{bmatrix}2&3\\3&-4\end{bmatrix}\begin{bmatrix}1&0\\-1&0\end{bmatrix}$
$=\begin{bmatrix}2-3&0\\3+4&0\end{bmatrix}=\begin{bmatrix}-1&0\\7&0\end{bmatrix}$
And $\text{A}(\text{BC})=\begin{bmatrix}1&2\\-2&1\end{bmatrix}\begin{bmatrix}-1&0\\7&0\end{bmatrix}$
$=\begin{bmatrix}-1+14&0\\2+7&0\end{bmatrix}=\begin{bmatrix}13&0\\9&0\end{bmatrix}\ ....(\text{ii})$
From (i) and (ii), we get
$\therefore\ (\text{AB})\text{C}=\text{A}(\text{BC})$

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

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{1-\sqrt{\text{xa}}}\Big)$
→
Evaluate:
$\int\sqrt{\tan\theta}\text{ d}\theta. $
→
A firm manufactures two products A and B. Each product is processed on two machines $M_1$ and $M_2$. Product A requires 4 minutes of processing time on $M_1$ and 8 min. on $M_2$; product B requires 4 minutes on $M_1$ and 4 min. on $M_2$. The machine $M_1$ is available for not more than 8 hrs 20 min. while machine $M_2$ is available for 10 hrs. during any working day. The products A and B are sold at a profit of Rs. 3 and Rs. 4 respectively.
Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.
→
Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}})+9=0.$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 6 - 9x - x^2$
→
Evalute the following integrals:
$\int\frac{\sec\text{x}}{\log(\sec\text{x}+\tan\text{x})}\text{dx}$
→
Differentiate the functions given in Exercise:
$\text{x}^\text{x}-2^{\sin\text{x}}$
→
Maximise $Z = 3x + 5y$
such that $\text{x}+3\text{y}\geq3,\ \text{x}+\text{y}\geq2,\ \text{x},\ \text{y}\geq0.$
→
Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum squares of its sides.
→
Evaluate the following integrals:$\int\text{x}^2\sin^{-1}\text{x dx}$
→