Question

The co-factors of the elements of second column of $\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]$ are

• A. -13, 6, 5
• B. 13, 5, 6
• C. 13, -6, -5
• D. -13, -6, 5

Hint:

Remember the alternating sign pattern for a $3 \times 3$ matrix: the second column signs are always $(-, +, -)$. Calculate the minors and simply apply these signs to get the cofactors quickly.

Answer 1

The Correct Option is A

Step 1: Recall the cofactor idea.
The cofactor of an element is the minor (the smaller determinant left after deleting that element's row and column) times a sign $(-1)^{i+j}$.

Step 2: Identify the second column.
The matrix is $\begin{bmatrix}1 & -1 & 2\\ 3 & 2 & 1\\ -1 & 3 & 4\end{bmatrix}$. The second column elements are $-1$ (row 1), $2$ (row 2), $3$ (row 3).

Step 3: Cofactor of the $(1,2)$ element.
Delete row 1 and column 2: $\begin{vmatrix}3 & 1\\ -1 & 4\end{vmatrix} = 12 + 1 = 13$. Sign is $(-1)^{1+2} = -1$. So cofactor $= -13$.

Step 4: Cofactor of the $(2,2)$ element.
Delete row 2 and column 2: $\begin{vmatrix}1 & 2\\ -1 & 4\end{vmatrix} = 4 + 2 = 6$. Sign is $(-1)^{2+2} = +1$. So cofactor $= 6$.

Step 5: Cofactor of the $(3,2)$ element.
Delete row 3 and column 2: $\begin{vmatrix}1 & 2\\ 3 & 1\end{vmatrix} = 1 - 6 = -5$. Sign is $(-1)^{3+2} = -1$. So cofactor $= 5$.

Step 6: Collect the answers.
The cofactors are $-13, 6, 5$. \[ \boxed{-13,\ 6,\ 5 \text{ (Option 1)}} \]