If $A =\begin{bmatrix}5&6\\3&2\end{bmatrix}$ then which of the following is correct?
(A) $|A|$ is positive
(B) $|adj\, A| = -8$
(C) Cofactor of 3 is 6
(D) $|2A| = -32$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (D) only **
$A=\begin{bmatrix}5 & 6 \\ 3 & 2\end{bmatrix}$
Determinant:
$|A| = 5\cdot2 - 6\cdot3 = 10 - 18 = -8$
(A) $|A|$ is positive → false (it is $-8$)
(B) $|\text{adj}\,A| = -8$
For a $2\times2$ matrix: $|\text{adj}\,A| = |A|^{1} = -8$ → true
(C) Cofactor of $3$:
Cofactor of $3 = (-1)^{2+1}\cdot\text{minor} = -\left|\begin{matrix}6\end{matrix}\right| = -6$ → not $6$ → false
(D) $|2A|$:
$|2A| = 2^{2}|A| = 4(-8) = -32$ → true
Correct statements: (B) and (D)