If A=100021103 then (A−I)(A−2I)(A−3I)=(I is the identity matrix of order 3)

A−I=100021103−100010001=000011102A−2I=100021103−200020002=−100000000 A−3I=100021103-300030000=−2000−11100 (A−I)(A−2I)(A−3I)=000011102 −100001101 −2000−11100 =0000111022+0+00+0+00+0+00+0+10+0+00+0+0−2+0+10+0+00+0+0=000011102200100−200=000000000=O

If A=100021103 then (A−I)(A−2I)(A−3I)=(I is the identity matrix of order 3)

A−I=100021103−100010001=000011102A−2I=100021103−200020002=−100000000 A−3I=100021103-300030000=−2000−11100 (A−I)(A−2I)(A−3I)=000011102 −100001101 −2000−11100 =0000111022+0+00+0+00+0+00+0+10+0+00+0+0−2+0+10+0+00+0+0=000011102200100−200=000000000=O

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If $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 3 \end{matrix}]$ then $(A - I) (A - 2 I) (A - 3 I) =$
(I is the identity matrix of order 3)

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Detailed Solution

$A - I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 3 \end{matrix}] - [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{matrix}]$

$A - 2 I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 3 \end{matrix}] - [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}] = [\begin{matrix} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] A - 3 I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 3 \end{matrix}] - [\begin{matrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 0 \end{matrix}] = [\begin{matrix} - 2 & 0 & 0 \\ 0 & - 1 & 1 \\ 1 & 0 & 0 \end{matrix}] (A - I) (A - 2 I) (A - 3 I) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{matrix}] [\begin{matrix} - 1 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{matrix}] [\begin{matrix} - 2 & 0 & 0 \\ 0 & - 1 & 1 \\ 1 & 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{matrix}] [\begin{matrix} 2 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 1 & 0 + 0 + 0 & 0 + 0 + 0 \\ - 2 + 0 + 1 & 0 + 0 + 0 & 0 + 0 + 0 \end{matrix}] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{matrix}] [\begin{matrix} 2 & 0 & 0 \\ 1 & 0 & 0 \\ - 2 & 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] = O$