Let $A=\begin{bmatrix}2&-3&4\\0&1&5\\-4&2&3\end{bmatrix}$, and $a_{ij}$ be any element of matrix A, $i, j ∈ \{1,2,3\}$, then which of the following are TRUE?
(A) Minor of $a_{23} = 16$
(B) Minor of $a_{23} = -8$
(C) Cofactor of $a_{23} = -16$
(D) Cofactor of $a_{23} = 8$
(E) Cofactor of $a_{13} = 4$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (D) and (E) only
$A=\begin{pmatrix}2&-3&4\\0&1&5\\-4&2&3\end{pmatrix}$
Minor of $a_{23}$
Delete row $2$ and column $3$
$M_{23}=\begin{vmatrix}2&-3\\-4&2\end{vmatrix}=2(2)-(-3)(-4)$
$=4-12=-8$
Hence
(B) is true, (A) is false
Cofactor of $a_{23}$
$A_{23}=(-1)^{2+3}M_{23}=-(-8)=8$
Hence
(D) is true, (C) is false
Cofactor of $a_{13}$
Delete row $1$ and column $3$
$M_{13}=\begin{vmatrix}0&1\\-4&2\end{vmatrix}=0(2)-1(-4)=4$
$A_{13}=(-1)^{1+3}M_{13}=4$
Hence
(E) is true
The correct options are (B), (D) and (E).