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.
Determinant elements can have subdeterminants {minor, determinant} containing elements that are not in same row and column. For determinant with rows "a b" and "c d", element-a minor is d, because a is in first row and column, and d is not in first row and not in first column. The smallest minor is one element.
Minors {signed minor} {cofactor}| can have sign. Sign depends on sum of element row and column positions. If element is in row and column whose sum is odd, sign is -1. If element is in row and column whose sum is even, sign is +1. For determinant |a b / c d|, a's signed minor is +d, because element a has row 1 and column 1, which sum to 2, which is even. b's signed minor is -c, because element b has row 1 and column 2, which sum to 3, which is odd. Therefore, determinant value is a*d - b*c.
Determinants can have squares, cubes, and higher powers {modulus, determinant}.
Determinants have a number of rows {rank, determinant}. Determinants have same number of columns, because they are square. Matrix rank is biggest non-zero-determinant number of rows.
To find determinant value, copy determinant {Sarrus rule} {rule of Sarrus}. Write all determinant rows, except last row, below copy. Multiply elements on each diagonal. Change sign on ascending diagonals, going down to right, but not sign on descending diagonals, going up to right. Add terms.
For a matrix representing linear homogeneous equations, partial-derivative coefficient determinant {Hessian determinant} indicates inflection points.
Determinants {Jacobian determinant} {functional determinant} can be for coordinate transformations: double integral of F(x,y) * dx * dy = double integral of G(u,v) * determinant(fu, fv, gu, gv) * du * dv. Jacobian determinants have two rows of partial derivatives, one row for F(x,y) and one row for G(u,v). Jacobian determines scalar for unit vectors. Jacobians can determine normal vectors at function intersections.
Product of determinant inverse and second determinant B and first determinant A calculates third determinant C {similar determinant}, which is similar to second determinant: A^-1 * B * A = C.
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Date Modified: 2022.0225