Consider the matrices and . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

Question

Consider the matrices $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}]$ and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}]$ . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

Smart Achievers · Accepted Answer

(34)

We have, $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}] \Rightarrow | A | = - 4 + 8 = 4$

and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}] \Rightarrow | B | = 9 - 9 = 0$

$∴ A^{- 1} = \frac{1}{| A |} adj (A) = \frac{1}{4} [\begin{matrix} - 2 & 2 \\ - 4 & 2 \end{matrix}]$

Now, PA = B

$\Rightarrow P = B A^{- 1}$

$= [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}] \times \frac{1}{4} [\begin{matrix} - 2 & 2 \\ - 4 & 2 \end{matrix}] = \frac{1}{4} [\begin{matrix} - 42 & 24 \\ - 14 & 8 \end{matrix}]$

Now, $A Q = B \Rightarrow Q = A^{- 1} B$

$= \frac{1}{4} [\begin{matrix} - 2 & 2 \\ - 4 & 2 \end{matrix}] [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}] = \frac{1}{4} [\begin{matrix} - 4 & 12 \\ - 10 & - 30 \end{matrix}]$

Now, sum of diagonal elements of 2(P + Q)

$= (- \frac{46}{4} - \frac{22}{4}) = 2 (- \frac{68}{4}) = - 34$

$∴$ Absolute value = 34.

Solution

Q.
Consider the matrices $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}]$ and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}]$ . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

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Solution

Q. Consider the matrices A=[2–24–2] and B=[3913]. If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

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Q.
Consider the matrices $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}]$ and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}]$ . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]