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Determinants by Elimination
Each elementary matrix operation corresponds to following rules:
- Interchanging two rows/columns of matrix changes sign of determinant
- Multiplying single row/column of matrix by scalar has effect of multiplying determinant by scalar
- Adding multiple of row/column to another does not change value of determinant
If is a triangular matrix, then . For example,
Let be a nonsingular matrix.
We define the adjoint matrix of as follows:
Where is the matrix created by substituting the cofactor for element .
By Lemma 2.2.1, , and
Assuming is nonsingular,
Let be a nonsingular matrix, and let . Let be the matrix obtained by replacing the th column of by . If is the unique solution to , then