cofactor expansion

For a 5×5 matrix

Use rows with most amount of zeros

For a 5×5 matrix Use rows with most amount of zeros

Example:

Consider the following 5x5 matrix:
\begin{bmatrix}
2 & -1 & 3 & 0 & 4 \
0 & 0 & -2 & 1 & 0 \
1 & 5 & 0 & 2 & -3 \
0 & 0 & 0 & 4 & -2 \
3 & 2 & 1 & -5 & 0
\end{bmatrix}

1. Identify Rows with Most Zeros: Row 4 has three zeros, which is the maximum number of zeros in any single row.

2. Choose the Row: Select Row 4 (0 0 0 4 -2).

3. Calculate Cofactors Along the Selected Row: Since we have three zeros, we only need to calculate two cofactors. The formula for the cofactor $C<em>{ij}$ is $(-1)^{i+j}M</em>{ij}$, where $M_{ij}$ is the minor of the element in the i-th row and j-th column.

4. Expand Along Row 4:

   $det(A) = 0 \cdot C<em>{41} + 0 \cdot C</em>{42} + 0 \cdot C<em>{43} + 4 \cdot C</em>{44} + (-2) \cdot C_{45}$

   $det(A) = 4 \cdot C<em>{44} - 2 \cdot C</em>{45}$

5. Compute Cofactors $C<em>{44}$ and $C</em>{45}$

   $C_{44} = (-1)^{4+4} \cdot det(\text{Submatrix formed by removing Row 4 and Column 4})$

   $C_{45} = (-1)^{4+5} \cdot det(\text{Submatrix formed by removing Row 4 and Column 5})$

6. Find the Submatrices:

   Submatrix for $C_{44}$:

   \begin{bmatrix}
   2 & -1 & 3 & 4 \
   0 & 0 & -2 & 0 \
   1 & 5 & 0 & -3 \
   3 & 2 & 1 & 0
   \end{bmatrix}

   Submatrix for $C_{45}$:

   $$\begin{bmatrix}
   2 & -1 & 3 & 0 \
   0 & 0 & -2 & 1 \
   1 & 5 & 0 & 2 \
   3 & 2