First slide

Operations of matrix

Question

If $A = [\begin{array}{ll} 0 0 \\ 0 1 \end{array}]$ then the matrix given by $B = I + A + A^{2} + \dots + A^{k}$ is

Moderate

A

$[\begin{array}{ll} k 1 \\ 1 k \end{array}]$
B

$[\begin{array}{cc} 1 & 0 \\ 0 & k + 1 \end{array}]$
C

$[\begin{array}{ll} 1 k \\ 0 1 \end{array}]$
D

None of these

Solution

$A^{k} = A for all k \Rightarrow B = [\begin{array}{cc} 1 & 0 \\ 0 & k + 1 \end{array}]$

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