Let P = [ 3 - 1 - 2 2 0 3 - 5 0 ] , where R . Suppose Q = [ q i j ] is a matrix satisfying P Q = k I 3 for some non-zero k R . If q 23 = - k 8 and | Q | = k 2 2 , then 2 + k 2 is equal to

Question

Let $P =$ $[\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}]$ , where $α \in R$ . Suppose $Q = [q_{i j}]$ is a matrix satisfying $P Q = k I_{3}$ for some non-zero $k \in R$ . If $q_{23} = - \frac{k}{8}$ and $| Q |$ $= \frac{k^{2}}{2}$ , then $α^{2} + k^{2}$ is equal to

Smart Achievers · Accepted Answer

(3)

$Given that P Q = k I$

$| P |$ $.$ $| Q |$ $= k^{3}$

$\Rightarrow$ $α \in R$ $= 2 k \neq 0 \Rightarrow P is an invertible matrix$

$P Q = k I_{3}$ $P Q = k I$ $∴$ $Q = k P^{- 1} I$ $| Q |$

$∴$ $Q = \frac{adj P}{2}$ $∵$ $q_{23} = - \frac{k}{8}$

$∴$ $- \frac{(3 α + 4)}{2} = \frac{k}{8} \Rightarrow 12 α + 16 = k$ $. . . (i)$

$∴$ $| P |$ $= 2 k \Rightarrow k = 10 + 6 α$

$From (i) and (ii) we get α = - 1, k = 4$

$∴$ $α^{2} + k^{2} = 17$

Solution

Q.
Let $P =$ $[\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}]$ , where $α \in R$ . Suppose $Q = [q_{i j}]$ is a matrix satisfying $P Q = k I_{3}$ for some non-zero $k \in R$ . If $q_{23} = - \frac{k}{8}$ and $| Q |$ $= \frac{k^{2}}{2}$ , then $α^{2} + k^{2}$ is equal to

1 13

2 15

3 17

4 21

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