Let P= [3&-1&-2 2&0&α 3&-5&0], where α ∈ R. Suppose Q = [qij] is a matrix satisfying PQ = kI3 for some non -zero k ∈ R If q23 = -(k/8) and |Q| = (k2/2), then α2 + k2 is equal to .

Question

Let P= [3&-1&-2 2&0&α 3&-5&0], where α ∈ R. Suppose Q = [qij] is a matrix satisfying PQ = kI3 for some non -zero k ∈ R If q23 = -(k/8) and |Q| = (k2/2), then α2 + k2 is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

PQ = kI | P | ⋅| Q |= k 3 ⇒ mid P| =2 k ≠ 0 ⇒ P is an invertible matrix ∵ PQ = kI ∴ Q = kP -1 I ∴ Q =( text adj.P /2) ∵ q 23=-( k /8) (-(3 α+4)/2)=-( k /8) ⇒ k =4 ∴ | P |=2 k ⇒ k =10+6 α ldots( i ) Put value of k in (i).. we get α=-1

Q. Let P=⎣⎡​323​−10−5​−2α0​⎦⎤​, where α∈R. Suppose Q=[qij​] is a matrix satisfying PQ=kI3​ for some non -zero k∈R If q23​=−8k​ and ∣Q∣=2k2​, then α2+k2 is equal to ________.

PQ=kI ∣P∣⋅∣Q∣=k3 ⇒∣P∣=2k=0 ⇒P is an invertible matrix ∵PQ=kI ∴Q=kP−1I ∴Q=2 adj.P ​ ∵q23​=−8k​ 2−(3α+4)​=−8k​ ⇒k=4 ∴∣P∣=2k ⇒k=10+6α…(i) Put value of k in (i).. we get α=−1

Q. Let $P = ⎣ ⎡ 323 - 1 0 - 5 - 2 α 0 ⎦ ⎤$ , where $α \in R$ . Suppose $Q = [q_{ij}]$ is a matrix satisfying $PQ = k I_{3}$ 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 ________.