"Let A be a 3 $\times$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :"

Question

Accepted Answer

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We know, $|adj\,A| = |A{|^{n - 1}}$

Now, $|adj\,24A| = |adj\,3(adj\,2A)|$

$\Rightarrow |24A{|^{3 - 1}} = |3\,adj\,2A{|^{3 - 1}}$

$\Rightarrow |24A{|^2} = |3\,adj\,2A{|^2}$

Also, we know, $|KA| = {K^n}|A|$

$$ \Rightarrow {\left( {{{(24)}^2}} \right)^2}|A{|^2} = {\left( {{{(3)}^3}} \right)^2}|adj\,2A{|^2}$$

$\Rightarrow {(24)^6}|A{|^2} = {3^6}\,.\,{\left( {|2A{|^{3 - 1}}} \right)^2}$

$\Rightarrow {(24)^6}|A{|^2} = {3^6}\,.\,|2A{|^4}$

$\Rightarrow {(24)^6}|A{|^2} = {3^6}\,.\,{\left( {{2^3}} \right)^4}\,.\,|A{|^4}$

$\Rightarrow {3^6}\,.\,{8^6}\,.\,|A{|^2} = {3^6}\,.\,{8^4}\,.\,|A{|^4}$

$\Rightarrow {8^2} = |A{|^2}$

$\Rightarrow |A{|^2} = 64$ = 2⁶

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Let A be a 3 $\times$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|² is equal to :