1. Tardigrade
  2. Question
  3. Mathematics
  4. 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 )

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Q. Let $Z=\begin{bmatrix} 1 & 1 & 3 \\ 5 & 1 & 2 \\ 3 & 1 & 0 \end{bmatrix}$ and $P=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 3 & 0 & 1 \end{bmatrix}$ . If $Z=PQ^{- 1}$ , where $Q$ is a square matrix of order $3,$ then the value of $Tr\left(\left(a d j Q\right) P\right)$ is equal to (where $Tr\left(\right.A\left.\right)$ 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 $B$ )

NTA AbhyasNTA Abhyas 2020Matrices

Solution:

Given, $Z=PQ^{- 1}$
$\left|P\right|=-5,\left|Z\right|=10$
$\left|Z\right|=\left|\frac{P}{Q}\right|\Rightarrow \left|Q\right|=\frac{- 5}{10}=-\frac{1}{2}$
$Z\left|Q\right|=P.adj\left(Q\right)$ $\Rightarrow P.adj\left(Q\right)=-\frac{1}{2}Z$
$Tr\left(\left(a d j Q\right) . P\right)=$ $Tr\left(P . a d j Q\right)$
$=\left(-\frac{1}{2} \operatorname{Tr}(Z)\right)$
$=-1$