Enter the matrix A, row and column indices i and j, and the element a_ij into the calculator to determine the determinant of a matrix using cofactors.
Cofactor Determinant Formula
The following formula is used to calculate the determinant of a matrix using cofactors.
Det(A) = Σ (-1)^{(i+j)} * a_ij * Det(M_ij)Variables:
- Det(A) is the determinant of matrix A
- i and j are the row and column indices of the element a_ij in matrix Aa_i
- j 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
To calculate the determinant using cofactors, for each element a_ij in the matrix, calculate the determinant of the sub-matrix that remains after removing the i-th row and j-th column. Multiply this by the element a_ij and by (-1) raised to the power of (i+j). Sum all these values to get the determinant of the matrix.
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.
How to Calculate Cofactor Determinant?
The following steps outline how to calculate the Cofactor Determinant using the given formula:
- First, identify the matrix A.
- Next, determine the value of i and j for each element a_ij in matrix A.
- For each element a_ij, calculate (-1)^(i+j) * a_ij * Det(M_ij).
- Sum up all the calculated values from step 3.
- The final result is the Cofactor Determinant of matrix A.
Example Problem:
Use the following variables as an example problem to test your knowledge:
Matrix A:
| 2 3 1 |
| 4 0 -2 |
| 1 -1 3 |
Calculate the Cofactor Determinant of matrix A.
