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Question:

Let P = ⎡⎢⎣3 2 3𕒵0 𕒹 𕒶α 0⎤⎥⎦, where α ∈ R. Suppose Q = [qij] is a matrix such that PQ = kI, where k ∈ R, k ≠ 0 and I is the identity matrix of order 3. If q23 = -k/8 and det(Q) = k2/2, then find the values of α and k.

4α - k + 8 = 0

det(Qadj(P)) = 213

α = 0, k = 8

det(Padj(Q)) = 29

Solution:

AsPQ=kI⇒Q=kP𕒵I|P|=⎡⎢⎣323𕒵0𕒹𕒶α0⎤⎥⎦,=(20+12α).. (1)NowQ=k|P|(adjP)I⇒Q=k(20+12α)⎡⎢⎣5α10−α3α0(𕒷α𕒸)𕒵0𕒵20⎤⎥⎦⎡⎢⎣100010001⎤⎥⎦∵q23=−k8⇒−k(3α+4)(20+12α)=−k8⇒2(3α+4)=5+3α3α=𕒷⇒α=𕒵Also|Q|=k3|I||P|⇒k22=k3(20+12α)∵from(1)