Example 1: Calculating the Determinant of a 2x2 Matrix
Matrix:
| 1 2 |
| 3 4 |
Steps:
Choose a row or column to expand. We can choose any row or column.
- For each element in the chosen row/column, calculate its minor. The minor of an element is the determinant of the submatrix obtained by deleting the row and column containing that element.
- Multiply each minor by its corresponding cofactor. The cofactor of an element is equal to its minor multiplied by (-1)^(i+j), where i and j are the indices of the element.
- Sum the products of minors and cofactors. This sum will be the determinant of the original matrix.
Calculation:
Expanding along row 1:
det(A) = 1 * C11 + 2 * C12
C11 = det([4]) = 4
C12 = -det([3]) = 3
det(A) = 1 * 4 + 2 * 3 = 10
Therefore, the determinant of the matrix is 10.
Example 2: Calculating the Determinant of a 3x3 Matrix
Matrix:
|1 3 2 |
| 4 5 1 |
| 2 6 7 |
Steps:
Choose a row/column to expand. Expanding along a row with more zeros can simplify the calculation.
Follow the same steps as in the 2x2 example, calculating minors and cofactors for each element in the chosen row/column.
Sum the products of minors and cofactors. This sum will be the determinant of the original matrix.
Calculation:
Expanding along row 1:
det(A) = 1 * C11 + 3 * C12 + 2 * C13
C11 = det([5 1]) = -1
C12 = -det([4 7]) = -25
C13 = det([4 5]) = 20
det(A) = 1 * (-1) + 3 * (-25) + 2 * 20 = -68
Therefore, the determinant of the matrix is -68.
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