# TAMU MATH 311 - Lecture 11 web (21 pages)

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## Lecture 11 web

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MATH 311 504 Topics in Applied Mathematics Lecture 11 Properties of determinants 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 not invertible Definition in low dimensions Definition det a a a b ad bc c d a11 a12 a13 a21 a22 a23 a11a22 a33 a12a23a31 a13a21a32 a31 a32 a33 a13a22a31 a12 a21a33 a11a23 a32 Examples 3 3 matrices 3 2 0 1 0 1 3 0 0 2 1 2 0 1 3 2 3 0 0 0 2 2 1 0 3 1 3 4 9 5 1 4 6 0 2 5 1 2 3 4 5 0 6 0 0 0 0 3 6 2 0 4 0 3 1 5 0 1 2 3 6 General definition The general definition of the determinant is quite complicated as there are no simple explicit formula There are several approaches to defining determinants Approach 1 original an explicit but very complicated formula Approach 2 axiomatic we formulate properties that the determinant should have Approach 3 inductive the determinant of an n n matrix is defined in terms of determinants of certain n 1 n 1 matrices Mn R the set of n n matrices with real entries Theorem There exists a unique function det Mn R R called the determinant with the following properties 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 det I 1 Corollary det A 0 if and only if the matrix A is not invertible 3 2 0 Example A 1 0 1 det A 2 3 0 In the previous lecture we have transformed the matrix A into the identity matrix using elementary row operations interchange the 1st row with the 2nd row add 3 times the 1st row to the 2nd row add 2 times the 1st row to the 3rd row multiply the 2nd row by 1 2 add 3 times the 2nd row to the 3rd row multiply the 3rd row by 2 5 add 3 2 times the 3rd row to the 2nd row add 1 times the 3rd row to the 1st row 3 2 0 Example A 1 0 1 det A 2 3 0 In the previous lecture we have transformed the matrix A into the identity matrix using elementary row operations These included two row multiplications by 1 2 and by 2 5 and one row exchange It follows that det I 12 52 det A 15 det A Hence det A 5 det I 5 Other properties of determinants If a matrix A has two det A 0 a1 a2 b1 b2 a1 a2 identical rows then a3 b3 0 a3 If a matrix A has two rows proportional then det A 0 a1 a2 a3 a1 a2 a3 b1 b2 b3 r b1 b2 b3 0 ra1 ra2 ra3 a1 a2 a3 Distributive law for rows Suppose that matrices A B C are identical except for the i th row and the i th row of C is the sum of the i th rows of A and B Then det A det B det C a1 a2 a3 a1 a2 a3 a1 a1 a2 a2 a3 a3 b1 b2 b3 b1 b2 b3 b1 b2 b3 c1 c2 c3 c1 c2 c3 c1 c2 c3 Adding a scalar multiple of one row to another row does not change the determinant of a matrix a1 rb1 a2 rb2 a3 rb3 b1 b2 b3 c1 c2 c3 a1 a2 a3 rb1 rb2 rb3 a1 a2 a3 b1 b2 b3 b1 b2 b3 b1 b2 b3 c1 c2 c3 c1 c2 c3 c1 c2 c3 Definition A square matrix A aij is called upper triangular if all entries below the main diagonal are zeros aij 0 whenever i j The determinant of an upper triangular matrix is equal to the product of its diagonal entries a11 a12 a13 0 a22 a23 a11a22a33 0 0 a33 If A diag d1 d2 dn then det A d1 d2 dn In particular det I 1 Definition Given a matrix A the transpose of A denoted AT or At is the matrix obtained by interchanging rows and columns in the matrix A That is if A aij then AT bij where bij aji T a1 b1 a1 a2 a3 a2 b2 Example b1 b2 b3 a3 b3 If A is a square matrix then det AT det A a1 a2 a3 a1 b1 c1 a2 b2 c2 b1 b2 b3 c1 c2 c3 a3 b3 c3 Columns vs rows If one column of a matrix is multiplied by a scalar the determinant is multiplied by the same scalar Interchanging two columns of a matrix changes the sign of its determinant If a matrix A has two columns proportional then det A 0 Adding a scalar multiple of one column to another does not change the determinant of a matrix Submatrices Definition Given a matrix A a k k submatrix of A is a matrix obtained by specifying k columns and k rows of A and deleting the other columns and rows 2 4 1 2 3 4 2 4 10 20 30 40 5 9 5 9 3 5 7 9 If A is an n n matrix then Aij denote the n 1 n 1 submatrix obtained by deleting the i th row and the jth column 3 2 0 Example A 1 0 1 2 3 0 0 1 1 1 1 0 A11 A12 A13 3 0 2 0 2 3 2 0 3 0 3 2 A21 A22 A23 3 0 2 0 2 3 2 0 3 0 3 2 A31 A32 A33 0 1 1 1 1 0 Row and column expansions Theorem Let A be an n n matrix Then for any 1 k m n we have that n X det A 1 k j akj det Akj j 1 expansion by kth row det A n X 1 i m aim det Aim i 1 expansion by mth column Signs for row column expansions 3 2 0 Example A 1 0 1 2 3 0 Expansion by the 1st row 3 2 0 0 1 1 1 1 0 1 3 2 5 3 0 2 0 2 3 0 Expansion by the 2nd row 3 2 0 3 2 2 0 5 1 1 0 1 1 2 3 3 0 2 3 0 3 2 0 Example A 1 0 1 2 3 0 Expansion by the 2nd column 3 2 0 1 1 3 0 1 0 1 2 3 5 2 0 1 1 2 3 0 Expansion by the 3rd column 3 2 0 3 2 5 1 0 1 1 2 …

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