दिया आव्यूह है
\(\mathrm{A}=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\)
\(A^2=A \times A\)
\( =\left[\begin{array}{lll}
1 \times 1+2 \times 2+2 \times 2 & 2 \times 1+1 \times 2+2 \times 2 & 2 \times 1+2 \times 2+1 \times 2 \\
2 \times 1+1 \times 2+2 \times 2 & 2 \times 2+1 \times 1+2 \times 2 & 2 \times 2+2 \times 1+2 \times 1 \\
1 \times 2+2 \times 2+1 \times 2 & 2 \times 2+1 \times 2+2 \times 1 & 2 \times 2+2 \times 2+1 \times 1
\end{array}\right]\)
\(\mathrm{A}^2=\left[\begin{array}{lll}9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9\end{array}\right]\) .....(i)
अब \(4 \mathrm{~A}=4\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\)
\(\mathrm{A}=\left[\begin{array}{lll}4 & 8 & 8 \\ 8 & 4 & 8 \\ 8 & 8 & 4\end{array}\right]\) ......(ii)
जैसा कि हम जानते हैं
\(\mathrm{I}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] ; \mathrm{5I}=\left[\begin{array}{ccc}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{array}\right]\) .....(iii)
चूँकि समीकरण (i), (ii) और (iii)
\(=\left[\begin{array}{lll}
9 & 8 & 8 \\
8 & 9 & 8 \\
8 & 8 & 9
\end{array}\right]-4\left[\begin{array}{lll}
4 & 8 & 8 \\
8 & 4 & 8 \\
8 & 8 & 4
\end{array}\right]-5\left[\begin{array}{lll}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{array}\right] \)
\(=\left[\begin{array}{lll}
9 & 8 & 8 \\
8 & 9 & 8 \\
8 & 8 & 9
\end{array}\right]-\left[\begin{array}{lll}
9 & 8 & 8 \\
8 & 9 & 8 \\
8 & 8 & 9
\end{array}\right]\)
\(=0\)
अतः \(\mathrm{A}^2-5 \mathrm{I}-4 \mathrm{A}=0\) सिद्ध हुआ।
