GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 5

Question

If $A =\begin{bmatrix}
1 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{bmatrix},$ the value of $(A^{-1})^{T}$ is _____
(Mark all the appropriate choices)

• A. $A$
• B. $Adj\;A$
• C. $\left(\left(A^{T}\right)^{T}\right)^{T}$
• D. $A^{T}$

Answer 1

Given that, $A = \begin{bmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix}$

We know that $A^{-1} = \dfrac{Adj\;A}{ \mid A \mid}$

$\mid A \mid  = -1$

$Adj\;A = \begin{bmatrix}
-1 & 0 & 0 \\
0 & 0 & -1 \\
0 & -1 & 0
\end{bmatrix}$

$\therefore A^{-1} = \begin{bmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix} = A$

Now, $(A^{-1})^{T} = A^{T}$

So, the correct answer is $A;C;D.$

Comments

I think adjoint matrix is wrong, a11 in adj(A) should be 1.