Cramer's Rule Cramer's Rule is a theorem that relates determinants of linear transformations and the determinants of subspaces. It provides a systematic way...
Cramer's Rule is a theorem that relates determinants of linear transformations and the determinants of subspaces. It provides a systematic way to compute the determinant of a matrix by evaluating determinants of smaller submatrices.
Theorem:
Let A be an m x n matrix and B be an m x k submatrix of A. Then:
det(A) = det(B)
Properties of Determinants:
Determinant of the identity matrix is the product of the determinants of the diagonal elements.
Determinant of a singular matrix (with zero determinant) is zero.
Determinant of a row echelon form matrix is the product of the determinants of the submatrix at that position.
Determinant of a diagonal matrix is the product of the determinants of the diagonal elements.
Determinant of a matrix is zero if and only if the matrix is singular.
Examples:
Determinant of the identity matrix is 1.
Determinant of a diagonal matrix with diagonal elements 1, 2, 3 is 6.
Determinant of a 2x2 matrix with entries 1, 2 and 3, 4 is 12.
Determinant of a 3x3 matrix with entries 1, 2, 3, 4, 5, 6, 7 is -24.
By understanding these properties and applying Cramer's Rule, we can efficiently compute determinants of matrices, both square and rectangular