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

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## Lecture 11 web

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- Math 311 - Top In Applied Math I

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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

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