Question

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

• A. $adj\left(adj\right(A\left)\right)=|A|\cdot \left(adj\right(A){)}^{-1}$
• B. $adj\left(adj\right(A\left)\right)=|A{|}^2\cdot \left(adj\right(A){)}^{-1}$
• C. $adj\left(adj\right(A\left)\right)=|A|\cdot A$
• D. $adj\left(A\right)=|A|\cdot A^{-1}$

Answer 1

The correct option is A $adj\left(adj\right(A\left)\right)=|A|\cdot \left(adj\right(A){)}^{-1}$

we know that $A.adjA=|A|I_2$

$adj\left(adj\right(A\left)\right)=|A{|}^{n-2}A=|A{|}^{3-2}A=|A|.A$ so 3rd is correct

using $adjA=|A|A^{-1}$.....4th is true

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

$=|A{|}^2(adjA{)}^{-1}$

so 2nd is true

we can say 1 st is wrong option