Find minors and cofactors of the elements $a_{11}$, $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix}$.

Question

Accepted Answer

> [!TLDR]
> The minor of an element is the determinant left after deleting its row and column; the cofactor is $(-1)^{i+j}$ times the minor. For $a_{11}$, minor $M_{11} = \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix}$ and cofactor $A_{11} = M_{11}$. For $a_{21}$, minor $M_{21} = \begin{vmatrix} a_{12} & a_{13} \ a_{32} & a_{33} \end{vmatrix}$ and cofactor $A_{21} = -M_{21}$.

The idea is simple: every element in a determinant has a "shadow" — the smaller determinant you get by removing its row and column. That shadow is the **minor**. But the sign matters when you use it to expand the determinant, so we attach a sign based on the element's position: $(-1)^{	ext{row}+	ext{column}}$. That signed minor is the **cofactor**.

Let’s walk through it for the two elements asked.

1. **For $a_{11}$ (first row, first column):**  
   Delete row 1 and column 1. What remains is the $2 	imes 2$ determinant formed by the other rows (2 and 3) and other columns (2 and 3):  
   $$M_{11} = \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix}$$  
   The cofactor $A_{11}$ is $(-1)^{1+1} M_{11} = (+1) \cdot M_{11} = M_{11}$.  
   So $A_{11} = \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix}$.

2. **For $a_{21}$ (second row, first column):**  
   Delete row 2 and column 1. The remaining rows are 1 and 3, remaining columns are 2 and 3:  
   $$M_{21} = \begin{vmatrix} a_{12} & a_{13} \ a_{32} & a_{33} \end{vmatrix}$$  
   The cofactor $A_{21} = (-1)^{2+1} M_{21} = (-1)^3 M_{21} = -M_{21}$.  
   So $A_{21} = -\begin{vmatrix} a_{12} & a_{13} \ a_{32} & a_{33} \end{vmatrix}$. …

Q.Find minors and cofactors of the elements $a_{11}$ , $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ .

Concept understanding — Determinant Evaluation Using Identities

Determinant Evaluation Using Identities

The geometric intuition

The three row (or column) operations

The strategy

Worked example

Method: Finding Minors and Cofactors in a General (Symbolic) Determinant

Steps

Common Mistakes

Why determinants change under row operations

Step-by-step evaluation

🎓Unlock everything free for 14 days

Unlock everything free for 14 days