MATH 323 Lecture 8

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Determinants by Elimination

Each elementary matrix operation corresponds to following rules:

  1. Interchanging two rows/columns of matrix changes sign of determinant
  2. Multiplying single row/column of matrix by scalar has effect of multiplying determinant by scalar
  3. Adding multiple of row/column to another does not change value of determinant

If is a triangular matrix, then . For example,

Cramer's Rule

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,


Formal Theorem

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