Download App

Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices5 Marks
Question
If $A=\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right], B=\left[\begin{array}{cc}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right]$ and $C=\left[\begin{array}{cc}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right]$

Show that

i. A+B=B+A

ii. (A + B) + C = A + (B + C)
✓

Answer

$A+B=\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]+\left[\begin{array}{cc}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right]$

$=\left[\begin{array}{cc}2-1 & -3+2 \\ 5+2 & -4+2 \\ -6+0 & 1+3\end{array}\right]$

$\therefore \quad A+B=\left[\begin{array}{cc}1 & -1 \\ 7 & -2 \\ -6 & 4\end{array}\right]$

....(i)

$B+A=\left[\begin{array}{cc}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right]+\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]$

$=\left[\begin{array}{cc}-1+2 & 2-3 \\ 2+5 & 2-4 \\ 0-6 & 3+1\end{array}\right]$

$\therefore \quad B+A=\left[\begin{array}{cc}1 & -1 \\ 7 & -2 \\ -6 & 4\end{array}\right]$

From (i) and (ii), we get A + B = B + A

$(A+B)+C=\left\{\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]+\left[\begin{array}{cc}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right]\right\}+\left[\begin{array}{cc}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right]$

$=\left[\begin{array}{cc}2-1 & -3+2 \\ 5+2 & -4+2 \\ -6+0 & 1+3\end{array}\right]+\left[\begin{array}{cc}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right]$

$=\left[\begin{array}{cc}1 & -1 \\ 7 & -2 \\ -6 & 4\end{array}\right]+\left[\begin{array}{cc}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right]$

$=\left[\begin{array}{cc}1+4 & -1+3 \\ 7-1 & -2+4 \\ -6-2 & 4+1\end{array}\right]$

$\therefore \quad(\mathrm{A}+\mathrm{B})+\mathrm{C}=\left[\begin{array}{cc}5 & 2 \\ 6 & 2 \\ -8 & 5\end{array}\right]$

$\ldots$...(i)

$\begin{aligned} A+(B+C) & =\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]+\left\{\left[\begin{array}{cc}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right]+\left[\begin{array}{cc}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right]\right\} \\ & =\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]+\left[\begin{array}{cc}-1+4 & 2+3 \\ 2-1 & 2+4 \\ 0-2 & 3+1\end{array}\right] \\ & =\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]+\left[\begin{array}{cc}3 & 5 \\ 1 & 6 \\ -2 & 4\end{array}\right]\end{aligned}$

$=\left[\begin{array}{cc}2+3 & -3+5 \\ 5+1 & -4+6 \\ -6-2 & 1+4\end{array}\right]$

$=\left[\begin{array}{cc}5 & 2 \\ 6 & 2 \\ -8 & 5\end{array}\right]$

..(ii)

$(A+B)+C=A+(B+C)$

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 "Solve the Following Question.(5 Marks)" from Determinants and Matrices→Determinants and Matrices — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the $20^{th}$ term and the sum of $20$ terms of the series:
$2 × 4 + 4 × 6 + 6 × 8 + ....$
→
If $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$, prove that $A^n=\left[\begin{array}{cc}1+2 n & -4 n \\ n & 1-2 n\end{array}\right]$, for all $n \in \mathbf{N}$.
→
If A = {2, 3}, B = {4, 5}, C = {5, 6}, find $\text{A}\times(\text{B}\cap\text{C}),\text{ A}\times(\text{B}\cap\text{C}),(\text{A}\times\text{B})\cup(\text{A}\times\text{C}).$
→
Find the equation of the hyperbola whose,
Focus is (2, 1) directrix is $2\text{x}+3\text{y}=1$ and eccentricity = 2
→
Two sides of an isosceles triangle are given by the equations $7x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, -10)$. Determine the equation of the third side.
→
For the function $\text{f}(\text{x})=\frac{\text{x}^{100}}{100}+\frac{\text{x}^{99}}{99}+\dots+\frac{\text{x}^2}{2}+\text{x}+1.$Prove that $\text{f}'(1)=100\text{f}'(0).$
→
Differentiate the following from the first principle$\cos\Big(\text{x}-\frac{\pi}{8}\Big)$
→
In $\triangle A B C$ prove that
$\frac{\cos A-\cos B+\cos C+1}{\cos A+\cos B+\cos C-1}=\cot \frac{A}{2} \cot \frac{C}{2}$
→
Let $\text{f}:[0,\infty)\rightarrow\text{R}$ and $\text{g}:\text{R}\rightarrow\text{R}$ be defined by $\text{f(x)}=\sqrt{\text{x}}$ and g(x) = x. Find f + g, g - g, fg and $\frac{\text{f}}{\text{g}}$
→
Show that $\text{x}^2+\text{xy}+\text{y}^2,\ \text{z}^2+\text{zx}+\text{x}^2$ and $\text{y}^2+\text{yz}+\text{z}^2$ are consecutive terms of an A.P., if x, y and z are in A.P.
→