Q. If $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix}$and $A^{- 1} = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ - 4 & 3 & c \\ \frac{5}{2} & \frac{1 - 3}{2} & \frac{1}{2} \end{bmatrix}$, then the values Of $a$ and $c$ are equal to
A) 1,1
B) 1,$- 1$
C) 1, 2
D) $- 1,\ 1$
Explanation
(b)
We have,
$I = \frac{1}{2}\begin{bmatrix}
0 & 1 & 2 \\
1 & 2 & 3 \\
3 & a & 1
\end{bmatrix}\begin{bmatrix}
1 & - 1 & 1 \\
- 08 & 6 & 2c \\
5 & - 3 & 1
\end{bmatrix}$
$= \begin{bmatrix}
1 & 0 & c + 1 \\
0 & 1 & 2(c + 1) \\
4(1 - a) & 3(a - 1) & 2 + ac
\end{bmatrix}$
Comparing the elements of $AA^{- 1}$ with those of $I$, we have
$c + 1 = 0 \Rightarrow c = - 1$
$\Rightarrow c = - 1\ and\ a - I = 0 \Rightarrow a = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.