Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015
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.
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.
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
Basic Theorems (2 of 2) Theorem Let A be a square matrix, then det(a) = det(a T ).
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.
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.
Elementary Row Operations Questions: what effect do elementary row operations have on the calculation of the determinant?
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).
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
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.
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 )
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.
Examples Let A = 5 4 1 1 2 4 and calculate the following determinants. 5 4 1 2 4 8 1 2 4 5 4 1 6 10 10 1 2 4 = = =
Examples Let A = 5 4 1 1 2 4 and calculate the following determinants. 5 4 1 2 4 8 1 2 4 5 4 1 6 10 10 1 2 4 = 92 = =
Examples Let A = 5 4 1 1 2 4 and calculate the following determinants. 5 4 1 2 4 8 1 2 4 5 4 1 6 10 10 1 2 4 = 92 = 46 =
Examples Let A = 5 4 1 1 2 4 and calculate the following determinants. 5 4 1 2 4 8 1 2 4 5 4 1 6 10 10 1 2 4 = 92 = 46 = 46
Elementary Matrices Recall: an elementary matrix results from applying a single elementary row operation to the identity matrix.
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) = 1. 3. 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.
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) = 1. 3. 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
Examples Evaluate the following determinants of elementary matrices. 1 0 0 0 0 3 0 0 0 0 1 0 = 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 = 0 0 0 1 1 0 0 2 0 1 0 0 0 0 1 0 = 0 0 0 1
Examples Evaluate the following determinants of elementary matrices. 1 0 0 0 0 3 0 0 0 0 1 0 = 3 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 = 0 0 0 1 1 0 0 2 0 1 0 0 0 0 1 0 = 0 0 0 1
Examples Evaluate the following determinants of elementary matrices. 1 0 0 0 0 3 0 0 0 0 1 0 = 3 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 = 1 0 0 0 1 1 0 0 2 0 1 0 0 0 0 1 0 = 0 0 0 1
Examples Evaluate the following determinants of elementary matrices. 1 0 0 0 0 3 0 0 0 0 1 0 = 3 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 = 1 0 0 0 1 1 0 0 2 0 1 0 0 0 0 1 0 = 1 0 0 0 1
Proportional Rows of Columns Theorem If A is an n n matrix with two proportional rows or columns, then det(a) = 0.
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.
Examples Find the determinants of the following matrices. 1 2 7 1. 8 8 4 4 4 2 4 7 4 6 2. 1 8 5 15 2 1 2 6 9 9 5 7 21 2
Row Reduction If we row-reduce a matrix to upper triangular form we can then easily calculate its determinant. Example Evaluate the following determinant. 0 6 6 6 3 3 6 3 4 2 2 0 2 4 0 0
Homework Read Section 2.2 Exercises: 1 29 odd