Let Z=[ 1 1 3 5 1 2 3 1 0 ] and P=[ 1 0 2 2 1 0 3 0 1 ] . If Z=PQ- 1 , where Q is a square matrix of order 3, then the value of Tr((a d j Q) P) is equal to (where Tr(.A.) represents the trace of a matrix A i.e. the sum of all the diagonal elements of the matrix A and adjB represents the adjoint matrix of matrix B )

Question

Let Z=[ 1 1 3 5 1 2 3 1 0 ] and P=[ 1 0 2 2 1 0 3 0 1 ] . If Z=PQ- 1 , where Q is a square matrix of order 3, then the value of Tr((a d j Q) P) is equal to (where Tr(.A.) represents the trace of a matrix A i.e. the sum of all the diagonal elements of the matrix A and adjB represents the adjoint matrix of matrix B ) (A) 3 (B) -1 (C) 4 (D) (6/5)

Tardigrade Admin · Accepted Answer

Given, Z=PQ- 1 |P|=-5,|Z|=10 |Z|=|(P/Q)|⇒ |Q|=(- 5/10)=-(1/2) Z|Q|=P.adj(Q) ⇒ P.adj(Q)=-(1/2)Z Tr((a d j Q) . P)= Tr(P . a d j Q) =(-(1/2) operatornameTr(Z)) =-1

$3$

$- 1$

$4$

$\frac{6}{5}$

3

−1

4

56​

Given, Z=PQ−1 ∣P∣=−5,∣Z∣=10 ∣Z∣=∣∣​QP​∣∣​⇒∣Q∣=10−5​=−21​ Z∣Q∣=P.adj(Q) ⇒P.adj(Q)=−21​Z Tr((adjQ).P)= Tr(P.adjQ) =(−21​Tr(Z)) =−1

$3$

$- 1$

$4$

$\frac{6}{5}$