Matrices define square element arrays {determinant, equation}|. Determinant symbol uses vertical lines at array sides: |A|. For square matrix, determinant elements are same as matrix elements. Second-order square matrices have rows "a b" and "c d": {a b / c d}, where / denotes row end. Determinant is |a b / c d|. Second-order square matrices have four elements. Third-order square matrices have nine elements. Fourth-order square matrices have 16 elements.
value
Determinants have scalar values, which are like area. To find determinant value, multiply each element of first column or first row by its signed minor. Add all products.
value: dependence
If a determinant row is a linear combination of other rows, determinant value equals zero.
value: triangular matrix
For triangular matrices, determinant value is product of principal-diagonal elements.
inverse
If matrix has determinant value zero, matrix is singular and has no inverse.
equation system
Equation systems have coefficient and constant arrays. Resultant determinant has variable coefficients: 2*x + 3*y = 0 and 4*x + 5*y = 0 goes to |2 3 / 4 5| = 2*5 + -3*4 = 2*5 + -4*3 = -2. Variables have determinants. Constants column replaces variable-coefficient column. For variable x, |0 3 / 0 5| = 0*5 + -3*0 = 0*5 + -0*3 = 0.
Determinative non-homogeneous linear-equation systems have determinant value not equal zero. Determinative homogeneous systems of linear equations have determinant value zero. To find variable values, use coefficient and constant determinant.
Mathematical Sciences>Algebra>Equation>System>Determinant
3-Algebra-Equation-System-Determinant
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Date Modified: 2022.0224