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प्रश्न
If\[A = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}\] f (x) = x2 − 2x − 3, show that f (A) = 0
उत्तर
Here,
`f (x) =x^2−2x−3`
`⇒f(A)=A^2−2A−3I_2`
Now,
`A^2=A A`
`⇒A^2=[[1 2],[2 1]] ,[[1 2],[2 1]]`
`⇒A^2= [[1+4 2+2],[2+2 4+1]]`
`⇒A^2= [[5 4],[4 5]]`
`f(A)=A^2−2A−3I_2`
⇒f(A)= `[[5 4],[4 5]]-2[[1 2],[2 1]]-3[[1 0],[0 1]]`
⇒f(A)=`[[5 4],[4 5]]-[[2 4],[4 2]]-[[3 0],[0 3]]`
⇒f(A)=`[[5-2-3 4-4-0],[4-4-0 5-2-3]]`
⇒f(A)= `[[5-5 0],[0 5 -5]]`
⇒f(A)=0
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