Let A be a n n matrix such that |A|=2.

Question

Let A be a $n	imes n$ matrix such that $|A|=2$. If the determinant of the matrix

$Adj(2\cdot Adj(2{A}^{-1}))\cdot$ is ${2}^{84}$, then $n$ is equal to :

Answer: 5

Accepted Answer

$\because |adj(2\cdot adj(2{A}^{-1}))|={2}^{84}$

$\Rightarrow {2}^{n\cdot (n-1)}{|adj(2{A}^{-1})|}^{(n-1)}={2}^{84}$

$\Rightarrow {2}^{n(n-1)}{|2{A}^{-1}|}^{(n-1{)}^{2}}={2}^{84}$

$\Rightarrow {2}^{n(n-1)}\cdot {2}^{n(n-1{)}^{2}}\cdot \frac{1}{|A{|}^{(n-1{)}^{2}}}={2}^{84}$

$\Rightarrow {2}^{n(n-1)+n(n-1{)}^{2}-(n-1{)}^{2}}={2}^{84}{\because |4|=2}$

$	herefore n(n-1)+(n-1{)}^{3}=84$

$	herefore n=5$

Solutions · 1