For the following matrices verify the distributivity of matrix, m… - Vidyadip

Question

For the following matrices verify the distributivity of matrix, multiplication over matrix addtion i.e., A(B + C) = AB + AC.
$\text{A}=\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix},\text{B}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}$

✓

Answer

$\text{A}(\text{B}+\text{C})=\text{AB}+\text{AC}$
$\Rightarrow\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{pmatrix}\begin{bmatrix}0&1\\1&1\end{bmatrix}+\begin{bmatrix}1&-1\\0&1\end{bmatrix}\end{pmatrix}$ $=\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}0&1\\1&1\end{bmatrix}+\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}1&-1\\0&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}0+1&1-1\\1+0&1+1\end{bmatrix}$ $=\begin{bmatrix}0-1&2-1\\0+1&1+1\\0+2&-1+2\end{bmatrix}+\begin{bmatrix}2-0&-2-1\\1+0&-1+1\\-1+0&1+2\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}1&0\\1&2\end{bmatrix}=\begin{bmatrix}-1&1\\1&2\\2&1\end{bmatrix}+\begin{bmatrix}2&-3\\1&0\\-1&3\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2-1&0-2\\1+1&0+2\\-1+2&0+4\end{bmatrix}=\begin{bmatrix}-1+2&1-3\\1+1&2+0\\2-1&1+3\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&-2\\2&2\\1&4\end{bmatrix}=\begin{bmatrix}1&-2\\2&2\\1&4\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$
Hence proved.

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

Prove that the area common to tha two parabola $y = 2x^2$ and $y = x^2 + 4$ is $323$ sq. units.

Find the equation of the normal lines to the curve $3x^2 - y^2 = 8$ which are parallel to the line $x + 3y = 4.$

Find the area of the region bounded by $y^2 = 9x, x = 2, x = 4$ and the x-axis in the first quadrant.

A small manufacturer has employed 5 skilled men and 10 semi-skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2 hrs. work by a skilled man and 2 hrs. work by a semi-skilled man. The ordinary model requires 1 hr by a skilled man and 3 hrs. by a semi-skilled man. By union rules no man may work more than 8 hrs per day. The manufacturers clear profit on deluxe model is Rs. 15 and on an ordinary model is Rs. 10. How many of each type should be made in order to maximize his total daily profit.

Give examples of two surjective functions $f_1$ and $f_2$ from $Z$ to $Z$ such that $f_1+f_2$ is not surjective.

If $\cos ^{-1} \frac{x}{a}+\cos ^{-1} \frac{y}{b}=\alpha$ then prove that$
\frac{x^2}{a^2}-\frac{2 x y}{a b} \cos \alpha+\frac{y^2}{b^2}=\sin ^2 \alpha
$

Evaluate the following integrals:
$\int_{0}^\limits{1}\text{x}\tan^{-1}\text{x}\text{ dx} $

Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$

Evaluate the following integrals:
$\int\limits^{1}_0\big|\text{x}\sin\pi\text{x}\big|\text{dx}$