Let A be any 3 [ ] 3 invertible matrix.

Question

Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?

Accepted Answer

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.

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