# if A is an invertible matrix of order 3 and |A|=5 the find the value of |adjA|

Consider,
${A}^{-1}=\frac{adj\left(A\right)}{\left|A\right|}\phantom{\rule{0ex}{0ex}}⇒A×{A}^{-1}=A×\frac{adj\left(A\right)}{\left|A\right|}\phantom{\rule{0ex}{0ex}}⇒I=A×\frac{adj\left(A\right)}{\left|A\right|}$

Since, A is invertible

$⇒I×\left|A\right|=A×adj\left(A\right)\phantom{\rule{0ex}{0ex}}⇒\left|I×\left|A\right|\right|=\left|A×adj\left(A\right)\right|$

$⇒{\left|A\right|}^{3}=\left|A\right|×\left|adj\left(A\right)\right|$ [Since, A is an invertible matrix of order 3]

$⇒\left|adj\left(A\right)\right|={\left|A\right|}^{2}\phantom{\rule{0ex}{0ex}}⇒\left|adj\left(A\right)\right|={\left(5\right)}^{2}=25$

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