If A = 3 & 1 -1 & 2 , show that A2 - 5A + 7I = 0. - Vidyadip

Question

If $\text{A} = \begin{bmatrix}3 & 1 \\-1 & 2\end{bmatrix},$ show that A² - 5A + 7I = 0.

✓

Answer

Given: $\text{A} = \begin{bmatrix}3 & 1 \\-1 & 2\end{bmatrix},$

$\therefore \ \text{A}^{2}-5\text{A}+7\text{I}=\begin{bmatrix}3&1\\-1&2\end{bmatrix}\begin{bmatrix}3&1\\-1&2\end{bmatrix}-5\begin{bmatrix}3&1\\-1&2\end{bmatrix}+7\begin{bmatrix}1&0\\0&1\end{bmatrix}$

$= \begin{bmatrix}9-1&3 + 2\\-3-2&-1 + 4\end{bmatrix}-\begin{bmatrix}15 & 5 \\-5 & 10\end{bmatrix} + \begin{bmatrix}7 & 0 \\0 & 7 \end{bmatrix} $$= \begin{bmatrix}8 & 5 \\-5 & 3 \end{bmatrix} - \begin{bmatrix}15 & 5\\-5& 10 \end{bmatrix}+ \begin{bmatrix}7 & 0 \\0 & 7 \end{bmatrix}$

$=\begin{bmatrix}8-15&5-5\\-5+5&3-10\end{bmatrix}+\begin{bmatrix}7&0\\0&7\end{bmatrix}$$=\begin{bmatrix}-7&0\\0&-7\end{bmatrix}+\begin{bmatrix}7&0\\0&7\end{bmatrix}=\begin{bmatrix}-7+7&0+7\\0+7&-7+7\end{bmatrix}$

$=\begin{bmatrix}0&0\\0&0\end{bmatrix}=0 = \text{R.H.S.}$

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 "3 Marks" 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

Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

If $4\sin^{-1}\text{x}+\cos^{-1}\text{x}=\pi,$ then what is the value of x?

Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then,
If c₁= 1 and c₂= 2, find c₃which makes $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

Let f be an invertible real function. Write (f^-1of)(1) + (f^-1of)(2) + ..... + (f^-1of)(100).

The probability of a shooter hitting a target is $\frac{3}{4}$. How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99 ?

A fair die is rolled. Consider events E = $\{1,\ 3,\ 5\},\ \text{F}=\{2,\ 3\}\ \text{and}\ \text{G}=\{2,\ 3,\ 4,\ 5\}.\ \text{Find}:$

$\text{P}\Big[(\text{E}\cup\text{F})|\text{G}\Big]\ \text{and}\ \text{P}\Big[(\text{E}\cap\text{F})|\text{G}\Big]$

Find the area of the parallelogram determinrd by the vectors:
$2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}$ and $\hat{\text{i}}-\hat{\text{j}}$

Differentiate the following w.r.t. x:
$\cos\big(\tan\sqrt{\text{x}+1}\big)$

If getting 5 or 6 in a throw of an unbiased die is a success and the random variable X denotes the number of successes in six throws of the die, find $\text{P}(\text{X}\geq4).$