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

Sum of all elements of P = 5 , Let a i j denote the elements of matrix P . Then, a 11 × a 22 = - 2

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

Let P = a b c d ; P 2 − 3 = − 11 6 ⇒ 2 a − 3 b = − 11 Also P − 11 − 6 = − 7 − 12 ⇒ − 11 c − 6 d = − 12 − 11 a − 6 d = − 7 ⇒ a = − 1 , b = 3 , c = 0 , d = 2

Let a i j denote the elements of matrix P . Then, a 11 × a 22 = - 2

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

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$Sum of all elements of P = 5$

$P [_{3}^{2}] = [_{6}^{7}]$

P is a nil-potent matrix.

$Let a_{i j} denote the elements of matrix P . Then, a_{11} \times a_{22} = - 2$

answer is A, D.

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Detailed Solution

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