Download App

Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]3 Marks
Question
If $A =\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]$, show that $A ^{-1}=\frac{1}{6}( A -5 I )$.
✓

Answer

$ |A|=\left|\begin{array}{ll} 4 & 5 \\ 2 & 1 \end{array}\right|=4-10=-6 \neq 0$
$\therefore A ^{-1}$ exists.
Consider $AA ^{-1}=1$
$\therefore\left[\begin{array}{ll} 4 & 5 \\2 & 1 \end{array}\right] A ^{-1}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1
\end{array}\right]$
By $\left(\frac{1}{4}\right) R_1$, we get,
$ \left[\begin{array}{ll} 1 & \frac{5}{4} \\ 2 & 1 \end{array}\right] A ^{-1}=\left[\begin{array}{ll} \frac{1}{4} & 0 \\
0 & 1 \end{array}\right] $
By $R_2-2 R_1$ we get,
$ \left[\begin{array}{cc} 1 & \frac{5}{4} \\ 0 & -\frac{3}{2} \end{array}\right] A ^{-1}=\left[\begin{array}{cc} \frac{1}{4} & 0 \\ -\frac{1}{2} & 1 \end{array}\right]$
By $\left(-\frac{2}{3}\right) R _2$, we get,
$ {\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] A ^{-1}=\left[\begin{array}{cc} -\frac{1}{6} & \frac{5}{6} \\ \frac{1}{3} & -\frac{2}{3}\ \end{array}\right]} $
$ \therefore A ^{-1}=\frac{1}{6}\left[\begin{array}{cc} -1 & 5 \\ 2 & -4 \end{array}\right] \ldots .(1) $
$ \frac{1}{6}( A -5 I )=\frac{1}{6}\left\{\left[\begin{array}{ll} 4 & 5 \\ 2 & 1 \end{array}\right]-5\left[\begin{array}{ll}
1 & 0 \\ 0 & 1 \end{array}\right]\right\} $
$ =\frac{1}{6}\left\{\left[\begin{array}{ll} 4 & 5 \\ 2 & 1 \end{array}\right]-\left[\begin{array}{ll} 5 & 0 \\ 0 & 5
\end{array}\right]\right\} $
$ =\frac{1}{6}\left[\begin{array}{cc} -1 & 5 \\ 2 & -4 \end{array}\right]$
From $(1)$ and $(2)$, we get $A ^{-1}=\frac{1}{6}( A -5 I )$

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.(3 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $e^{x+y} - x = 0$, prove that $\frac{\text{dy}}{\text{dx}}=\frac{1-\text{x}}{\text{x}}$
→
Check the commutativity and associativity of the following binary operations:
$'⊙'$ on $Q$ defined by $a ⊙ b = a^2 + b^2$  for all $a, b \in Q.$
→
Write the interval in which $\text{f}(\text{x})=\sin\text{x}+\cos\text{x},\text{x}\in\Big[0,\frac{\pi}{2}\Big]$ is increasing.
→
Write a value of $\int\text{e}^{3\log\text{x}}\text{x}^{4}\text{ dx}.$
→
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as six appears on at least one die
→
Find the differential equation by eliminating arbitrary constants from the relation $y = (c_1 + c_2x)e^x$
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}-1&4\\2&3 \end{vmatrix}$
→
Evaluate the following integrals:
$\int\cos^7\text{x}\text{ dx}$
→
Evalute the following integrals:
$\int\frac{\cot\text{x}}{\log\sin\text{x}}\text{dx}$
→
Represent the following families of curves by forming the corresponding differential equation:
$\text{y}=4\text{ax}$
→