If A=0300 and f(x)=x+x2+x3+....+x2023 then f(A)+I=

A=0300∵A2=0000then An=0,n≥2∵f(x)=x+x2+….+x2023f(A)=A+A2+…..+A2023f(A)=Af(A)+I=A+If(A)+I=0300+1001=1301

If A=0300 and f(x)=x+x2+x3+....+x2023 then f(A)+I=

A=0300∵A2=0000then An=0,n≥2∵f(x)=x+x2+….+x2023f(A)=A+A2+…..+A2023f(A)=Af(A)+I=A+If(A)+I=0300+1001=1301

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If $A = [\begin{matrix} 0 & 3 \\ 0 & 0 \end{matrix}]$ and $f (x) = x + x^{2} + x^{3} + .... + x^{2023}$ then $f (A) + I =$

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$[\begin{matrix} 1 & 3 \\ 1 & 1 \end{matrix}]$

$[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

$[\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}]$

$[\begin{matrix} 1 & 3 \\ 0 & 0 \end{matrix}]$

answer is C.

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Detailed Solution

$\begin{array}{l} A = [\begin{array}{ll} 0 & 3 \\ 0 & 0 \end{array}] \\ ∵ A^{2} = [\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}] \end{array}$
then $A^{n} = 0, n \geq 2$
$\begin{array}{l} ∵ f (x) = x + x^{2} + \dots . + x^{2023} \\ f (A) = A + A^{2} + \dots . . + A^{2023} \\ f (A) = A \\ f (A) + I = A + I \\ f (A) + I = [\begin{array}{ll} 0 & 3 \\ 0 & 0 \end{array}] + [\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}] \\ = [\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}] \end{array}$