How to Calculate Determinant (DET) of a Matrix
The determinant of a matrix is a single numerical value that can be used to determine various properties of the matrix, such as whether it is invertible or singular. It is often denoted by the symbol “det”.
Methods for Calculating DET
- 2×2 Matrix: For a 2×2 matrix [[a b] [c d]], DET = (a * d) – (b * c).
- 3×3 Matrix (Laplace Expansion): For a 3×3 matrix, the determinant can be calculated using Laplace expansion. Select any row or column as the “pivot” and use the corresponding minors and cofactors to calculate the DET.
- Higher-Order Matrices (Gauss-Jordan Elimination): For matrices of order greater than 3, Gauss-Jordan elimination can be used to reduce the matrix to an upper triangular matrix, and the DET can be easily calculated as the product of the diagonal elements.
Formula for 3×3 Matrix (Laplace Expansion)
Consider a 3×3 matrix A = [[a11 a12 a13] [a21 a22 a23] [a31 a32 a33]]. The determinant can be calculated using Laplace expansion along the first row:
DET(A) = a11 * C11 – a12 * C12 + a13 * C13
where Cij is the cofactor of aij, calculated as:
Cij = (-1)^(i+j) * M(ij)
where M(ij) is the minor of aij, which is the determinant of the 2×2 matrix obtained by deleting row i and column j from A.
Examples
Example 1 (2×2 Matrix):
Calculate DET for [[2 3] [4 5]].
DET = (2 * 5) – (3 * 4) = 1
Example 2 (3×3 Matrix using Laplace Expansion):
Calculate DET for [[1 2 3] [4 5 6] [7 8 9]].
DET = a11 * C11 – a12 * C12 + a13 * C13
= 1 * (3 * 9 – 6 * 8) – 2 * (4 * 9 – 6 * 7) + 3 * (4 * 8 – 5 * 7)
= 9
Conclusion
The determinant of a matrix is a useful mathematical tool that can be applied in various areas such as linear algebra, geometry, and engineering. Understanding its calculation methods and formulas is essential for solving complex matrix problems.
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