How to Solve question ? If A is a square matrix of order 3 such that A(Adj(A))=[-2 0 0;0 -2 0;0 0 -2]then find |A|
Solution :
Given that A is a square matrix of order 3 such that \[A(Adj(A))=\left[\begin{matrix}-2&0&0\\0&-2&0\\0&0&-2\\\end{matrix}\right]\] to find |A| Since We know that \[A^{-1}=\frac{Adj\left(A\right)}{\left|A\right|}\] Multiplying by A both sides to get \[AA^{-1}=\frac{A\left(Adj\left(A\right)\right)}{\left|A\right|}\] \[I=\frac{A\left(Adj\left(A\right)\right)}{\left|A\right|}\] Now apply determinant both sides \[\left|I\right|=\frac{\left|A\left(Adj\left(A\right)\right)\right|}{\left|\left|A\right|\right|}\] Since determinant of identity matrix is 1. \[1=\frac{\left|\begin{matrix}-2&0&0\\0&-2&0\\0&0&-2\\\end{matrix}\right|}{\left|A\right|}\] \[1=-\frac{8}{\left|A\right|}\] \[\left|A\right|=-8\] Therefore |A| is -8.