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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 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Determinant, Matrices, elementary row operations, Special Matrices Determinants, Row Operations Determinant

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