If A= 1&2 2&1 , f(x) = x^2 - 2x - 3, show that f(A) = 0 - Vidyadip

Question

If $\text{A}=\begin{bmatrix}1&2\\2&1\end{bmatrix},$ $f(x) = x^2 - 2x - 3$, show that $f(A) = 0$

✓

Answer

Given: $\text{A}=\begin{bmatrix}1&2\\2&1\end{bmatrix}$ and$ f(x) = x^2 - 2x - 3$
$\text{f(A)}=\text{A}^2-2\text{A}-3\text{I}$
$=\begin{bmatrix}1&2\\2&1\end{bmatrix}\begin{bmatrix}1&2\\2&1\end{bmatrix}-2\begin{bmatrix}1&2\\2&1\end{bmatrix}-3\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$=\begin{bmatrix}1+4&2+2\\2+2&4+1\end{bmatrix}-\begin{bmatrix}2&4\\4&2\end{bmatrix}-\begin{bmatrix}3&0\\0&3\end{bmatrix}$
$ =\begin{bmatrix}5&4\\4&5\end{bmatrix}-\begin{bmatrix}2&4\\4&2\end{bmatrix}-\begin{bmatrix}3&0\\0&3\end{bmatrix}$
$=\begin{bmatrix}5-2-3&4-4-0\\4-4-0&5-2-3\end{bmatrix}$
$=\begin{bmatrix}0&0\\0&0\end{bmatrix}$
$=0$
So,
$\text{f(A)}=0$

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 Algebra of Matrices→Algebra of Matrices — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}=\text{at}^2,\text{y}=2\text{at at}\text{ t}=1$

Integrate the following w. r. t. x:

$\frac{(3 \sin x-2) \cdot \cos x}{5-4 \sin x-\cos ^2 x}$

Evaluate the following definite integrals:$\int_{0}^\limits{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^2+ 2x - 5$

A firm manufactures two types of products $A$ and $B$ and sells them at a profit of $R s 2$ on type $A$ and $R s 3$ on type $B$. Each product is processed on two machines $M_1$ and $M_2$. Type $A$ requires one minute of processing time on $M_1$ and two minutes of $M _2$; type $B$ requires one minute on $M _1$ and one minute on $M _2$. The machine $M _1$ is available for not more than 6 hours 40 minutes while machine $M _2$ is available for 10 hours during any working day. Formulate the problem as a LPP.

Find the absolute maximum and minimum values of a function f given by
$f(x) = 2x^3 - 15x^2+ 36x + 1$ on the interval $[1,5]$

Prove that:
$\begin{vmatrix}1&\text{b}+\text{c}&\text{b}^2+\text{c}^2\\1&\text{c}+\text{a}&\text{c}^2+\text{a}^2\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2 \end{vmatrix}=(\text{a}+\text{b})(\text{b}-\text{c})(\text{c}-\text{a})$

Evaluate the following integrals:$\int^\limits\frac{\pi}{2}_{0}\frac{\cos^2\text{x}}{1+3\sin^3\text{x}}\text{ dx}$

Find $\frac{d^2 y}{d x^2}$ if, : $x=\cot ^{-1}\left(\frac{\sqrt{1-t^2}}{t}\right)$ and $x=\operatorname{cosec}^{-1}\left(\frac{1+t^2}{2 t}\right)$

Evaluate the following integrals:
$\int(4\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$