Question A.2: Find the characteristic equation of the matrix A = |1 4 2 3 ......
Find the characteristic equation of the matrix
\overline{A}=\left|\begin{matrix} 1 & 4 & 2 \\ 3 & 2 & -2 \\ 1 & -1 & 2 \end{matrix} \right|
and then the inverse of the matrix.
The characteristic equation is given by
\left|\begin{matrix} 1-λ & 4 & 2 \\ 3 & 2-λ & -2 \\ 1 & -1 & 2-λ \end{matrix} \right|=0
Expanding, the characteristic equation is
λ³ – 5λ² – 8λ + 40 = 0
and then by the Cayley–Hamilton theorem
\overline{A}^2 -5\overline{A} -8\overline{I }+40 \overline{A}^{-1}=0
40 \overline{A}^{-1}=-\overline{A}^2 +5\overline{A} +8\overline{I }
We can write
40 A^{-1}=-\left|\begin{matrix} 1 & 4 & 2 \\ 3 & 2 & -2 \\ 1 & -1 & 2 \end{matrix} \right|^2 +5\left|\begin{matrix} 1 & 4 & 2 \\ 3 & 2 & -2 \\ 1 & -1 & 2 \end{matrix} \right| +8\left|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right|
The inverse is
A^{-1}=-\left|\begin{matrix} -0.05&0.25&0.3 \\ 0.2& 0 &-0.2 \\ 0.125 & -0.125 & 0.25 \end{matrix} \right|
This is not an effective method of finding the inverse for matrices of large dimensions.