Statement-I: The matrix 1−3−4−1341−3−4 is a nilpotent matrix Statement-II : The matrix 1 1 35 3 6−2 −1 −3 is an idempotent matrix Which of the above Statement(s) is true?

A
only I is true
B
only II true
C
Both I & II are true
D
Neither I Nor II are true

Solution:

$A = [\begin{array}{ccc} 1 & - 3 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 3 & - 4 \end{array}] If A^{k} = 0, then A is nilpotent of index k . \begin{array}{l} A^{2} = [\begin{array}{ccc} 1 & - 3 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 3 & - 4 \end{array}] [\begin{array}{ccc} 1 & - 3 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 3 & - 4 \end{array}] \\ = [\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] \\ A^{2} = 0 ∴ A is nilpotent with index 2 . \end{array}$
$A = [\begin{array}{rrr} 1 1 3 \\ 5 3 6 \\ - 2 - 1 - 3 \end{array}] A^{2} = [\begin{array}{rrr} 1 1 3 \\ 5 3 6 \\ - 2 - 1 - 3 \end{array}] [\begin{array}{rrr} 1 1 3 \\ 5 3 6 \\ - 2 - 1 - 3 \end{array}] = [\begin{matrix} 0 & 1 & 0 \\ 8 & 8 & 15 \\ - 1 & - 2 & - 3 \end{matrix}] A^{4} = [\begin{matrix} 0 & 1 & 0 \\ 8 & 8 & 15 \\ - 1 & - 2 & - 3 \end{matrix}] [\begin{matrix} 0 & 1 & 0 \\ 8 & 8 & 15 \\ - 1 & - 2 & - 3 \end{matrix}] = [\begin{matrix} 8 & 8 & 15 \\ 49 & 42 & 105 \\ - 13 & - 11 & - 21 \end{matrix}]$
A is not nilpotent matrix

$Statement-I: The matrix [\begin{array}{ccc} 1 & - 3 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 3 & - 4 \end{array}] is a nilpotent matrix$
$Statement-II : The matrix [\begin{array}{rrr} 1 1 3 \\ 5 3 6 \\ - 2 - 1 - 3 \end{array}] is an idempotent matrix$
$Which of the above Statement(s) is true?$

Solution:

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