If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right]$, then which of the following are incorrect?
A. $|A|=5$
B. $Adj~A=\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right]$
C. $A(Adj~A)=25 I$
D. $|3 B|=45$
E. $(A B)^{-1}=B^{-1} A^{-1}$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) - B, C only
$|A|=2×4-3=5$
$|B|=3×1+(-2)×1=5$
$|3B|=3^2×|B|=45$ (A, D correct)
for A → cofactors
$C_{11}=4, C_{12}=-1$
$C_{21}=-3,C_{22}=2$
$Adj\,A=\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right]^T=\left[\begin{array}{ll}4 & -3 \\ -1 & 2\end{array}\right]$ (B incorrect)
also $A(adj\,A)=5I$ (C incorrect)
B already incorrect