QJEEM17Ma1f7e: JEE Main 2017 Mathematics - Class 12 JEEM17Ma1f7e Solution: Let A be any 3 3 invertible matrix. Then which one of the…

We know, the formula

A^-1 = ${{adj\left( A \right)} \over {\left| A \right|}}$

$\therefore$ adj (A) = $\left| \right.$ A $\left| \right.$ .A ^{$-$ 1}

So, Option (A) is true.

We know, the formula

adj (adj (A)) = ${\left| A \right|^{n - 2}}.A$

Now if we put n = 3 as given that A is a 3 $\times$ 3 matrix, we get

adj (adj (A)) = ${\left| A \right|^{3 - 2}}.A$ = $\left| A \right|.A$

So, Option (B) is also true.

We know, the formula

adj (adj (A)) = ${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$

Now if we put n = 3 as given that A is a 3 $\times$ 3 matrix, we get

adj (adj (A)) = ${\left| A \right|^{3 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$ = ${\left| A \right|^{2}}{\left( {adj\left( A \right)} \right)^{ - 1}}$

So, Option (C) is also true.

Now in this formula

adj (adj (A)) = ${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$

if we put n = 2, we get

adj (adj (A)) = ${\left| A \right|^{2 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$ = ${\left| A \right|}{\left( {adj\left( A \right)} \right)^{ - 1}}$

But as A is a 3 $\times$ 3 matrix so we can not take n = 2, so we can say for a 3 $\times$ 3 matrix option (D) is not true.

So, Option (D) is false.