Matrices Question 14
Question 14 - 31 January - Shift 2
Let $A$ be a $n \times n$ matrix such that $|A|=2$. If the determinant of the matrix $Adj(2 Adj(2 A^{-1}).$ ). is $2^{84}$, then $n$ is equal to ___________
Show Answer
Answer: 5
Solution:
Formula: Properties of Adjoint of a Matrix, Determinant properties, Properties of Inverse of a matrix
$|Adj(2 Adj(2 A^{-1}))|$
$=|2 Adj(Adj(2 A^{-1}))|^{n-1}$
$=2^{n(n-1)}|Adj(2 A^{-1})|^{n-1}$
$=2^{n(n-1)}|(2 A^{-1})|^{(n-1)(n-1)}$
$=2^{n(n-1)} 2^{(n-1)(n-1)}|A^{-1}|^{(n-1)(n-1)}$
$=2^{n(n-1)+n(n-1)(n-1)} \frac{1}{|A|^{(n-1)^{2}}}$
$=\frac{2^{n(n-1)+n(n-1)(n-1)}}{2^{(n-1)^{2}}}$
$=2^{n(n-1)+n(n+1)^{2}-(n-1)^{2}}$
$=2^{(n-1)(n^{2}-n+1)}$
Now, $2^{(n-1)(n^{2}-n+1)}$
$2^{(n-1)(n^{2}-n+1)}=2^{84}$
So, $n=5$
