If $A=\begin{bmatrix}5&2\\4&3\end{bmatrix}$ is a given matrix, then which of the following statements are correct?
(A) $|A|=7$
(B) minor of $3=-5$
(C) co-factor of $2=-4$
(D) $adj(A) =\begin{bmatrix}3&-2\\-4&5\end{bmatrix}$
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only **
$A=\begin{bmatrix}5&2\\4&3\end{bmatrix}$
(A) $|A|=5\cdot3-2\cdot4=15-8=7$ → True
(B) Minor of element $3$ is the determinant of the submatrix left after removing its row & column → minor = $5$ (not $-5$) → False
(C) Cofactor of $2$:
Minor of $2$ = $4$
Position of $2$ is $(1,2)$ → sign $=(-1)^{1+2}=-1$
Cofactor = $-4$ → True
(D) $\text{adj}(A)=\begin{bmatrix}3&-2\\-4&5\end{bmatrix}$ → True
Correct statements: (A), (C), (D)