Question

If a = [aij]3 x 3, such that , then 1 + log1/2det[adj(adj a)] is equal to

Answer 1

$\textbf{Given:}$

$\text{A is a square matrix of order 3}$

$\textbf{To find:}$

$\text{The value of }\; 1+\log_{\frac{1}{2}}det[adj(adjA)] }$

$\textbf{Solution:}$

$\text{We know that,}$

$\textbf{If A is a square matrix  of order n, then}$

$\bf\,det[adj(adjA)]=[det(A)]^{(n-1)^2}$

$\text{Consider,}$

$1+\log_{\frac{1}{2}}det[adj(adjA)]$

$=1+\log_{\frac{1}{2}}[det(A)]^{(n-1)^2}$

$=1+\log_{\frac{1}{2}}[det(A)]^{(3-1)^2}$

$=1+\log_{\frac{1}{2}}[det(A)]^{2^2}$

$=1+\log_{\frac{1}{2}}[det(A)]^4$

$=1+4\,\log_{\frac{1}{2}}det(A)$

$\textbf{Answer:}$

$\bf\,1+\log_{\frac{1}{2}}det[adj(adjA)]=1+4\,\log_{\frac{1}{2}}det(A)$