Question: The multiplicative inverse of matrix \[\left[ {\begin{array}{*{20}{c}} 2&1 \\\ 7&4 \end{...

Question

The multiplicative inverse of matrix $\left[ {\begin{array}{*{20}{c}} 2&1 \\\ 7&4 \end{array}} \right]$ is
A. $\left[ {\begin{array}{*{20}{c}} 4&{ - 1} \\\ 7&2 \end{array}} \right]$
B. $\left[ {\begin{array}{*{20}{c}} 4&{ - 1} \\\ { - 7}&2 \end{array}} \right]$
C. $\left[ {\begin{array}{*{20}{c}} 4&{ - 1} \\\ { - 7}&{ - 2} \end{array}} \right]$
D. $\left[ {\begin{array}{*{20}{c}} { - 4}&{ - 1} \\\ 7&{ - 2} \end{array}} \right]$

Answer 1

Solution

First, we will use the formula of the inverse of the matrix $A = \left[ {\begin{array}{*{20}{c}} a&b; \\\ c&d; \end{array}} \right]$ by, ${A^{ - 1}} = \dfrac{1}{{\left| A \right|}}\left[ {\begin{array}{*{20}{c}} d&{ - b} \\\ { - c}&a; \end{array}} \right]$ , where $\left| A \right|$ is the determinant $A$ . Then we will find the value of $a$ , $b$ , $c$ and $d$ from the given matrix $A$ and then substitute them in the formula of inverse of matrix to find the required value.

Complete step by step answer:

We are given that the matrix is $\left[ {\begin{array}{*{20}{c}} 2&1 \\\ 7&4 \end{array}} \right]$ .
We know that the inverse of the matrix $A = \left[ {\begin{array}{*{20}{c}} a&b; \\\ c&d; \end{array}} \right]$ by using the formula, ${A^{ - 1}} = \dfrac{1}{{\left| A \right|}}\left[ {\begin{array}{*{20}{c}} d&{ - b} \\\ { - c}&a; \end{array}} \right]$ , where $\left| A \right|$ is the determinant of $A$ .

Finding the value of $a$ , $b$ , $c$ and $d$ from the given matrix $A$ , we get
$\Rightarrow a = 2$
$\Rightarrow b = 1$
$\Rightarrow c = 7$
$\Rightarrow d = 4$

Then we will compute the value of determinant of $A$ using the above values, we get

\Rightarrow \left| A \right| = \left| {\begin{array}{*{20}{c}} 2&1 \\\ 7&4 \end{array}} \right| \\\ \Rightarrow \left| A \right| = 8 - 7 \\\ \Rightarrow \left| A \right| = 1 \\\

Substituting the above values of the determinant of A, $a$ , $b$ , $c$ and $d$ in the formula of inverse of matrix, we get

\Rightarrow {A^{ - 1}} = \dfrac{1}{1}\left[ {\begin{array}{*{20}{c}} 4&{ - 1} \\\ { - 7}&2 \end{array}} \right] \\\ \Rightarrow {A^{ - 1}} = \left[ {\begin{array}{*{20}{c}} 4&{ - 1} \\\ { - 7}&2 \end{array}} \right] \\\

Hence, option B is correct.

Note: In these types of questions, the key concept is to find the inverse by putting the values in the formula of inverse. Students should know that the matrix $\left[ {\begin{array}{*{20}{c}} d&{ - b} \\\ { - c}&a; \end{array}} \right]$ is the adjoint matrix of $A$ . We can remember this matrix to save some time in the $2 \times 2$ matrix but we have to compute the value of $adjA$ in the matrix more than 2 rows and 2 columns. When a student knows the formula of inverse, the solution is very simple and easy.