Let P be a matrix satisfying P2-3=-11-6 and P22-3=-7-12 , then which of the following is/are INCORRECT?

A
$Sum of all elements of P = 5$
B
$Let a_{i j} denote the elements of matrix P . Then, a_{11} \times a_{22} = - 2$
C
$P [_{3}^{2}] = [_{6}^{7}]$
D
P is a nil-potent matrix.

Solution:

$\begin{array}{l} Let \\ P = [\begin{array}{ll} a & b \\ c & d \end{array}]; P [\begin{array}{r} 2 \\ - 3 \end{array}] = [\begin{array}{r} - 11 \\ 6 \end{array}] \Rightarrow 2 a - 3 b = - 11 \\ Also P [\begin{array}{c} - 11 \\ - 6 \end{array}] = [\begin{array}{c} - 7 \\ - 12 \end{array}] \Rightarrow_{- 11 c - 6 d = - 12}^{- 11 a - 6 d = - 7} \\ \Rightarrow a = - 1, b = 3, c = 0, d = 2 \end{array}$

$Let P be a matrix satisfying P [\begin{matrix} 2 \\ - 3 \end{matrix}] = [\begin{matrix} - 11 \\ - 6 \end{matrix}] and P^{2} [\begin{matrix} 2 \\ - 3 \end{matrix}] = [\begin{matrix} - 7 \\ - 12 \end{matrix}], then which of the$ following is/are INCORRECT?

Solution:

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