Evaluating Determinants by Row Reduction

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1 Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015

2 Objectives Reduce a matrix to row echelon form and evaluate its determinant. Row reduction requires fewer operations than cofactor expansion and is therefore more efficient for large matrices.

3 Basic Theorems (1 of 2) Theorem Let A be a square matrix, if A has a row or column of zeros, then det(a) = 0.

4 Basic Theorems (1 of 2) Theorem Let A be a square matrix, if A has a row or column of zeros, then det(a) = 0. Proof. Suppose the ith row contains only zeros and perform cofactor expansion along the ith row. det(a) = (0)C i1 + (0)C i2 + + (0)C in = 0

5 Basic Theorems (2 of 2) Theorem Let A be a square matrix, then det(a) = det(a T ).

6 Basic Theorems (2 of 2) Theorem Let A be a square matrix, then det(a) = det(a T ). Proof. The ith row of A is the ith column of A T. Therefore cofactor expansion along the ith row of A is the same as cofactor expansion along the i column of A T.

7 Basic Theorems (2 of 2) Theorem Let A be a square matrix, then det(a) = det(a T ). Proof. The ith row of A is the ith column of A T. Therefore cofactor expansion along the ith row of A is the same as cofactor expansion along the i column of A T. Remark: rows vs. columns are irrelevant when talking about determinants.

8 Elementary Row Operations Questions: what effect do elementary row operations have on the calculation of the determinant?

9 Elementary Row Operations Questions: what effect do elementary row operations have on the calculation of the determinant? Theorem Let A be an n n matrix. 1. If B is the matrix that results when a single row or column of A is multiplied by a scalar k, then det(b) = k det(a). 2. If B is the matrix that results when two rows or columns of A are interchanged, then det(b) = det(a). 3. If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column is added to another column, then det(b) = det(a).

10 Illustrations for 3 3 Matrices ka 11 ka 12 ka 13 a 21 a 22 a 23 a 31 a 32 a 33 a 21 a 22 a 23 a 11 a 12 a 13 a 31 a 32 a 33 a 11 + ka 21 a 12 + ka 22 a 13 + ka 23 a 21 a 22 a 23 a 31 a 32 a 33 = k = = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

11 Proof (1) Suppose matrix B is the same as matrix A except that the ith row of B is the ith row of A multiplied by k.

12 Proof (1) Suppose matrix B is the same as matrix A except that the ith row of B is the ith row of A multiplied by k. Form the cofactor expansion of det(b) along the ith row. det(b) = ka i1 C i1 + ka i2 C i2 + + ka in C in = k(a i1 C i1 + a i2 C i2 + + a in C in )

13 Proof (1) Suppose matrix B is the same as matrix A except that the ith row of B is the ith row of A multiplied by k. Form the cofactor expansion of det(b) along the ith row. det(b) = ka i1 C i1 + ka i2 C i2 + + ka in C in = k(a i1 C i1 + a i2 C i2 + + a in C in ) = k det(a) since the cofactors C ij are independent of the entries in the i row.

14 Examples Let A = and calculate the following determinants = = =

15 Examples Let A = and calculate the following determinants = 92 = =

16 Examples Let A = and calculate the following determinants = 92 = 46 =

17 Examples Let A = and calculate the following determinants = 92 = 46 = 46

18 Elementary Matrices Recall: an elementary matrix results from applying a single elementary row operation to the identity matrix.

19 Elementary Matrices Recall: an elementary matrix results from applying a single elementary row operation to the identity matrix. Theorem Let E be an n n elementary matrix. 1. If E is the matrix that results from multiplying a row of I n by a scalar k, then det(e) = k. 2. If E is the matrix that results when two rows I n are interchanged, then det(e) = If E is the matrix that results when a multiple of one row of I n is added to another row, then det(e) = 1.

20 Elementary Matrices Recall: an elementary matrix results from applying a single elementary row operation to the identity matrix. Theorem Let E be an n n elementary matrix. 1. If E is the matrix that results from multiplying a row of I n by a scalar k, then det(e) = k. 2. If E is the matrix that results when two rows I n are interchanged, then det(e) = If E is the matrix that results when a multiple of one row of I n is added to another row, then det(e) = 1. Proof. det(i n ) = 1

21 Examples Evaluate the following determinants of elementary matrices = = =

22 Examples Evaluate the following determinants of elementary matrices = = =

23 Examples Evaluate the following determinants of elementary matrices = = =

24 Examples Evaluate the following determinants of elementary matrices = = =

25 Proportional Rows of Columns Theorem If A is an n n matrix with two proportional rows or columns, then det(a) = 0.

26 Proportional Rows of Columns Theorem If A is an n n matrix with two proportional rows or columns, then det(a) = 0. Proof. If the ith row is k times jth row, then multiply the jth row j by k and add to the ith row. This elementary row operation does not change the determinant of A. Now A has a row of zeros and thus det(a) = 0.

27 Examples Find the determinants of the following matrices

28 Row Reduction If we row-reduce a matrix to upper triangular form we can then easily calculate its determinant. Example Evaluate the following determinant

29 Homework Read Section 2.2 Exercises: 1 29 odd

Properties of the Determinant Function

Properties of the Determinant Function MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Overview Today s discussion will illuminate some of the properties of the determinant: