JAM 2022 [ 41 -50]

Let A be a 3 x 3 real matrix such that det(A) = 6 and

adj A = \left( 
\begin{matrix}
1 & -1 & 2 \\
5 & 7 & 1 \\
-1 & 1 & 1
\end{matrix}
\right)

where adj A denotes the adjoint of A.

Then the trace of A equals ________________________ (round off to 2 decimal places)

adj (adj A) = \left( \begin{matrix}
6 & -6  & 12 \\
3 & 3 & 0 \\
-15 & 9 & 12
\end{matrix} \right) ^T \\
\text{Since, A is a 3x3 matrix, we have,} \\
A = (A^{-1}) ^{-1}=\frac{adj(A^{-1})}{det(A^{-1})} = \frac{1}{det(A)}(adj (adj A)) \\
tr(A) = \frac{21}{6}= \frac{7}{2}=3.5

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