The formula to calculate the determinant of a matrix using cofactors is:
\[ \text{Det}(A) = \sum (-1)^{(i+j)} a_{ij} \text{Det}(M_{ij}) \]
Where:
A cofactor determinant is a mathematical concept used in the calculation of the determinant of a square matrix. It involves the use of cofactors, which are sub-matrices derived from the original matrix by removing one row and one column. The determinant of a matrix is calculated by multiplying each element of a row or column by the determinant of its cofactor and then summing these products, with alternating signs. This process can be recursively applied to calculate the determinants of larger matrices.
Example 1:
1, 2 3, 4
Step 1: Calculate the determinant:
\[ \text{Det}(A) = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2 \]
Example 2:
2, 3, 1 4, 1, 5 7, 2, 6
Step 1: Calculate the determinant:
\[ \text{Det}(A) = 2 \cdot \text{Det}\begin{vmatrix}1 & 5 \\ 2 & 6\end{vmatrix} - 3 \cdot \text{Det}\begin{vmatrix}4 & 5 \\ 7 & 6\end{vmatrix} + 1 \cdot \text{Det}\begin{vmatrix}4 & 1 \\ 7 & 2\end{vmatrix} \]
\[ = 2 \cdot (1 \cdot 6 - 5 \cdot 2) - 3 \cdot (4 \cdot 6 - 5 \cdot 7) + 1 \cdot (4 \cdot 2 - 1 \cdot 7) \]
\[ = 2 \cdot (6 - 10) - 3 \cdot (24 - 35) + 1 \cdot (8 - 7) \]
\[ = 2 \cdot (-4) - 3 \cdot (-11) + 1 \cdot 1 \]
\[ = -8 + 33 + 1 = 26 \]
