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Chapter 2.2 Computing Determinants through RowReduction

# Example prove the theorem for 2 2 matrices outline

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Unformatted text preview: ., column), then det B = det A. Example Prove the theorem for 2 × 2 matrices. Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix An Algorithm for Computing the Determinant This theorem provides an eﬃcient algorithm for computing the determinant of a matrix A: 1 2 3 Use elementary row operations to reduce A to a matrix B whose determinant is easy to compute (e.g., triangular). Keep track of the number of row swaps and the scalar multiplications. If the reduction to B used m row swaps and multiplication by the scalars k1 , k2 , . . . , kt , then −− det A = (−1)m k1 1 k2 1 · · · kt−1 det B . Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix Example Example 1 2 Compute the determinant of 1 1 A= 1 1 the matrix: 1 1 −1 −1 11 4 4 2 −2 8 −8 Compute the determinant of the matrix using column operations. 1 0 0 −3 −3 5 0 9 A= 0 −4 2 0 6 −1 7 8 Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix Example Example Compute the determinant of the matrix using a combination of cofactor expansion and elementary row operations: 2 −3 5 3 3 2 6 2 A= 1 4 −4 −2 4 −5 8 7 Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix Proportional Rows and Columns Corollary If A is an n × n matrix with two proportional row (or columns), then det A = 0. Proof. Apply a row operation to reduce A to a matrix with a row (or column) of zeros. Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix The Determinant of an Elementary Matrix Corollary Let E be an n × n elementary matrix. 1 If E is obtained by interchanging two rows of In , then det E = −1. 2 If E is obtained by multiplying a row of In by a scalar k (k = 0), then det E = k. 3 If E is obtained by adding a scalar multiple of one row to another row of In , then det E = 1....
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