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l_notes33

# l_notes33 - Lecture XXXIII Determinants Matrix Algebra 1...

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Lecture XXXIII Determinants; Matrix Algebra Determinants For a square matrix A , the determinant of A has the following properties: 1. Interchanging two rows of the matrix multiplies the value of the determi- nant by -1. 2. If there exists two identical rows, then the value of the determinant is 0. 3. If we multiply a row by a scalar c , then the value of the determinant is also multiplied by c . 4. Applying the β operation to the matrix leaves the value of the determinant unchanged. 5. If the determinant is 0, then there exists a row that can be written as a linear combination of the other rows. 6. Properties (1) through (5) also hold for columns instead of rows. In addition to the Laplace expansion method of ±nding determinants, there is a second method that is faster in most cases. This is a simpli±ed row-reduction that brings the matrix in to a form called row-echelon matrix. In this method we use operations β and γ . The row-echelon matrix is diﬀerent from the row- reduced matrix in that: 1. pivots can have other values than 1, since we do not use α operations; 2. in the column of a pivot, we only need to get 0’s below the pivot. 1

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l_notes33 - Lecture XXXIII Determinants Matrix Algebra 1...

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