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Question: If A = \(\begin{bmatrix} 0 & 5 \\ 0 & 0 \end{bmatrix}\) and f(x) = 1 + x + x<sup>2</sup>+ ..... + x<...

If A = [0500]\begin{bmatrix} 0 & 5 \\ 0 & 0 \end{bmatrix} and f(x) = 1 + x + x2+ ..... + x16, then f(1) is equal to –

A

0

B

[1501]\begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix}

C

[1500]\begin{bmatrix} 1 & 5 \\ 0 & 0 \end{bmatrix}

D

[0511]\begin{bmatrix} 0 & 5 \\ 1 & 1 \end{bmatrix}

Answer

[1501]\begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix}

Explanation

Solution

f(1) = I + A + A2 + ........ + A16

A = [0500]\begin{bmatrix} 0 & 5 \\ 0 & 0 \end{bmatrix} ̃ A2 = [0500][0500]\begin{bmatrix} 0 & 5 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 5 \\ 0 & 0 \end{bmatrix} = [0000]\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

A3 = A2.A = [0000]\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Similarly A4 = A5 =............. = A16 = [0000]\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

f(1) = [1001]+[0500]+[0000]+....+[0000]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 5 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + .... + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}= [1501]\begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix}