Question:medium

If $A = \left[\begin{array}{ccc}5 & 6 & 3 \\ -4 & 3 & 2 \\ -4 & -7 & 3\end{array}\right]$, then cofactors of all elements of second row are respectively

Show Hint

You can save a lot of time by using process of elimination! As soon as you compute the very first cofactor $C_{21} = -39$, you can immediately narrow your choices down to options (A) and (B). Then, checking just the middle sign or calculating $C_{22} = 27$ isolates the correct option without needing to calculate the final term!
Updated On: Jun 12, 2026
  • $-39, 3, 11$
  • $-39, 27, 11$
  • $39, -3, -11$
  • $39, -27, 11$
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Recall the cofactor rule.
The cofactor of the entry in row $i$, column $j$ is $C_{ij}=(-1)^{i+j}M_{ij}$, where $M_{ij}$ is the $2\times2$ determinant left after deleting that row and column.
Step 2: Note the sign pattern for row 2.
For $i=2$ the signs are $(-1)^{3}=-,\ (-1)^{4}=+,\ (-1)^{5}=-$, i.e. minus, plus, minus across the three entries.
Step 3: Cofactor of the first entry $a_{21}=-4$.
Delete row 2 and column 1: $\begin{vmatrix}6&3\\-7&3\end{vmatrix}=18-(-21)=39$. With the minus sign, $C_{21}=-39$.
Step 4: Cofactor of the second entry $a_{22}=3$.
Delete row 2 and column 2: $\begin{vmatrix}5&3\\-4&3\end{vmatrix}=15-(-12)=27$. With the plus sign, $C_{22}=27$.
Step 5: Cofactor of the third entry $a_{23}=2$.
Delete row 2 and column 3: $\begin{vmatrix}5&6\\-4&-7\end{vmatrix}=-35-(-24)=-11$. With the minus sign, $C_{23}=-(-11)=11$.
Step 6: Collect the cofactors.
In order the cofactors of the second row are $-39,\ 27,\ 11$, matching option (2).
\[ \boxed{-39,\ 27,\ 11} \]
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