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, $n=5$