Determinant Calculator

Formula

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:

Det(A) is the determinant of matrix A
i and j are the row and column indices of the element \(a_{ij}\) in matrix A
a_{ij} is an element in matrix A
Det(M_{ij}) is the determinant of the sub-matrix that remains after removing the i-th row and j-th column from matrix A

What is a Cofactor Determinant?

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 Calculation

Example 1:

Matrix A:

1, 2
3, 4

Step 1: Calculate the determinant:

\[ \text{Det}(A) = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2 \]

Example 2:

Matrix A:

2, 3, 1
4, 1, 5
7, 2, 6

Step 1: Calculate the determinant:

\[ \text{Det}(A) = 2 \cdot \text{Det}\begin{vmatrix}1 & 5 \\ 2 & 6\end{vmatrix} - 3 \cdot \text{Det}\begin{vmatrix}4 & 5 \\ 7 & 6\end{vmatrix} + 1 \cdot \text{Det}\begin{vmatrix}4 & 1 \\ 7 & 2\end{vmatrix} \]

\[ = 2 \cdot (1 \cdot 6 - 5 \cdot 2) - 3 \cdot (4 \cdot 6 - 5 \cdot 7) + 1 \cdot (4 \cdot 2 - 1 \cdot 7) \]

\[ = 2 \cdot (6 - 10) - 3 \cdot (24 - 35) + 1 \cdot (8 - 7) \]

\[ = 2 \cdot (-4) - 3 \cdot (-11) + 1 \cdot 1 \]

\[ = -8 + 33 + 1 = 26 \]