MATH 304 Linear Algebra Lecture 7 Evaluation of determinants Cramer s rule Determinant is a scalar assigned to each square matrix Notation The determinant of a matrix A aij 1 i j n is denoted det A or a11 a12 a21 a22 an1 an2 a1n a2n ann Principal property det A 0 if and only if the matrix A is singular Explicit definition in low dimensions Definition det a a a b ad bc c d a11 a12 a13 a21 a22 a23 a11 a22 a33 a12 a23 a31 a13 a21 a32 a31 a32 a33 a13 a22 a31 a12 a21 a33 a11 a23 a32 Properties of determinants Determinants and elementary row operations if a row of a matrix is multiplied by a scalar r the determinant is also multiplied by r if we add a row of a matrix multiplied by a scalar to another row the determinant remains the same if we interchange two rows of a matrix the determinant changes its sign Properties of determinants Tests for singularity if a matrix A has a zero row then det A 0 if a matrix A has two identical rows then det A 0 if a matrix has two proportional rows then det A 0 Properties of determinants Special matrices det I 1 the determinant of a diagonal matrix is equal to the product of its diagonal entries the determinant of an upper triangular matrix is equal to the product of its diagonal entries Properties of determinants Determinant of the transpose If A is a square matrix then det AT det A Columns vs rows if one column of a matrix is multiplied by a scalar the determinant is multiplied by the same scalar adding a scalar multiple of one column to another does not change the determinant interchanging two columns of a matrix changes the sign of its determinant if a matrix A has a zero column or two proportional columns then det A 0 Row and column expansions Given an n n matrix A aij let Mij denote the n 1 n 1 submatrix obtained by deleting the ith row and the jth column of A Theorem For any 1 k m n we have that n X det A 1 k j akj det Mkj j 1 expansion by kth row det A n X 1 i m aim det Mim i 1 expansion by mth column Signs for row column expansions 1 2 3 Example A 4 5 6 7 8 9 Expansion by the 1st row 2 3 1 5 6 4 6 4 5 7 8 8 9 7 9 det A 1 5 6 4 6 4 5 2 3 8 9 7 9 7 8 5 9 6 8 2 4 9 6 7 3 4 8 5 7 0 1 2 3 Example A 4 5 6 7 8 9 Expansion by the 2nd 1 2 4 6 5 7 7 9 det A 2 column 1 3 3 4 6 8 9 4 6 1 3 1 3 5 8 7 9 7 9 4 6 2 4 9 6 7 5 1 9 3 7 8 1 6 3 4 0 1 2 3 Example A 4 5 6 7 8 9 Subtract the 1st row from the 2nd row and from the 3rd row 1 2 3 1 2 3 1 2 3 4 5 6 3 3 3 3 3 3 0 7 8 9 7 8 9 6 6 6 since the last matrix has two proportional rows 1 2 3 Another example B 4 5 6 7 8 13 Let s do some row reduction Add 4 times the 1st row to the 2nd row 1 2 3 1 2 3 4 5 6 0 3 6 7 8 13 7 8 13 Add 7 times the 1st row to the 3rd row 1 2 3 1 2 3 0 3 6 0 3 6 0 6 8 7 8 13 1 2 3 1 2 3 0 3 6 0 3 6 0 6 8 7 8 13 Expand the determinant by the 1st column 1 2 3 3 6 0 3 6 1 6 8 0 6 8 Thus det B 3 2 3 6 1 2 3 6 8 6 8 1 2 3 2 2 12 3 4 2 2 0 3 5 3 2 1 Example C 1 1 0 3 det C 2 0 0 1 Expand the determinant by the 3rd column 2 2 0 3 2 2 3 5 3 2 1 2 1 1 3 1 1 0 3 2 0 1 2 0 0 1 Add 2 times the 2nd row to the 1st row 0 0 9 2 2 3 det C 2 1 1 3 2 1 1 3 2 0 1 2 0 1 2 2 3 0 0 9 det C 2 1 1 3 2 1 1 3 2 0 1 2 0 1 Expand the determinant by the 1st row 0 0 9 1 1 det C 2 1 1 3 2 9 2 0 2 0 1 Thus det C 18 1 1 18 2 36 2 0 Problem For what values of a will the following system have a unique solution x 2y z 1 x 4y 2z 2 2x 2y az 3 The system has a unique solution if and only if the coefficient matrix is invertible 1 2 1 A 1 4 2 det A 2 2 a 1 2 1 A 1 4 2 2 2 a Add 2 times the 1 2 1 4 2 2 det A 3rd column to the 2nd column 1 1 0 1 2 1 0 2 a 2 2 2a a Expand the determinant by the 2nd column 1 0 1 1 1 0 2 2 2a det A 1 1 2 2 2 2a a Hence det A 2 2a 3 6 1 a Thus A is invertible if and only if a 6 1 Properties of determinants Determinants and matrix multiplication if A and B are n n matrices then det AB det A det B if A and B are n n matrices then det AB det BA if A is an invertible matrix then det A 1 det A 1 Determinants and scalar multiplication if A is an n n matrix and r R then det rA r n det A Examples 1 2 1 1 0 0 X 0 2 2 Y 1 3 0 0 0 3 2 2 1 det X 1 2 3 6 det Y det Y T 3 det XY 6 3 18 det YX 3 6 18 det Y 1 1 3 det XY 1 6 3 2 det XYX 1 det Y 3 det X 1 Y 1 XY 1 det 2X 23 det X 23 6 48 det 3X T XY 4 3 3 6 6 3 4 12 A system of n linear equations in n variables a11 x1 a12 x2 a1n xn b1 a21 x1 a22 x2 a2n xn b2 Ax b an1 x1 an2 x2 ann xn bn where a11 a12 a …
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