la_m3_handouts

la_m3_handouts - MAC 2103 Module 3 Determinants 1 Learning...

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1 1 MAC 2103 Module 3 Determinants 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor expansion. 3. Determine the inverse of a matrix using the adjoint. 4. Solve a linear system using Cramer’s Rule. 5. Use row reduction to evaluate a determinant. 6. Use determinants to test for invertibility. 7. Find the eigenvalues and eigenvectors of a matrix. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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2 3 Rev.09 Determinants http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Determinants by Cofactor Expansion Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Evaluating Determinants by Row Reduction Properties of the Determinant Properties of the Determinant There are three major topics in this module: 4 Rev.F09 What is a Determinant? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Determinants are commonly used to test if a matrix is invertible and to find the area of certain geometric figures . A determinant is a real number associated with a square matrix. = a b c d
3 5 Rev.F09 How to Determine if a Matrix is Invertible? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The following is often used to determine if a square matrix is invertible. 6 Rev.F09 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Determine if A -1 exists by computing the determinant of the matrix A . a) b) Solution a) b) A -1 -1 does not exist does not exist A -1 does exist does exist det( A ) = 9 3 ! 3 ! 1 = (9)( ! 1) ! ( ! 3)(3) = 0 det( A ) = ! 5 9 4 ! 1 = ( ! 5)( ! ! (4)(9) = ! 31

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4 7 Rev.F09 What are Minors and Cofactors? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. We know we can find the determinants of 2 x 2 matrices; but can we find the determinants of 3 x 3 matrices, 4 x 4 matrices, 5 x 5 matrices, . ..? In order to find the determinants of larger square matrices, we need to understand the concept of minors and cofactors. 8 Rev.F09 Example of Finding Minors and Cofactors http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Find the minor M 11 and cofactor A 11 for matrix A . Solution To obtain M 11 begin by crossing out the first row and column of A . The minor is equal to det det B = B = 6(5) 6(5) ( ( 3)(7) 3)(7) = = 9 Since Since A 11 11 = ( = ( 1) 1) 1+1 1+1 M 11 11 , , A 11 can can be computed as follows: be computed as follows: A 11 = ( = ( 1) 1) 2 ( 9) = 9
5 9 Rev.F09 How to Find the Determinant of Any Square Matrix? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Once we know how to obtain a cofactor, we can find the determinant of any square matrix. You may pick any row or column, but the calculation is easier if some elements in the selected row or column equal 0.

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This note was uploaded on 10/18/2011 for the course MAS 2103 taught by Professor Shaw during the Summer '11 term at Valencia.

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la_m3_handouts - MAC 2103 Module 3 Determinants 1 Learning...

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