7_Matrices - The right half of the resulting augmented...

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Rank of a matrix 9/15/11 Rank of A is the size of the largest square submatrix of A whose determinant is nonzero. Rank of A is 2. All you need to show is there is a 2x2 submatrix of A that is nonsingular -- determinant is nonzero. Example:
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Inverse of a matrix Let A be a square matrix of size n. If |A| 0, then A is a non-singular matrix and there exists an n × n matrix, denoted A -1 , such that A.A -1 = I n . A -1 is unique. For a 2x2 matrix, the inverse is calculated as follows. Prove that AA -1 = A -1 A = I 2 and (A -1 ) -1 = A. Let us assume that |A| 0. Then, For larger square matrices, finding the inverse is significantly more complex.
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Elementary row operations Any nonsingular square matrix can be reduced to an identity matrix using elementary row operations. Elementary column operations can be defined in a similar manner.
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Gauss-Jordan elimination method to find the inverse of a matrix Use elementary row operations to reduce the left half of the augmented matrix to identity matrix.
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Unformatted text preview: The right half of the resulting augmented matrix is the inverse of the original matrix. Check: Example: Gauss-Jordan Elimination Method If the diagonal element of the current row, the element (i,i) position in iteration i, is 0, switch that row with another row that has nonzero value in that column position. For invertible matrices, this is always feasible. This switching of rows to ensure the diagonal element is nonzero is called pivoting . Solution to system of Linear equations Note : If the equations are independent, that is, none of the equations can be obtained by a linear combination of the other two equations, then the corresponding matrix is nonsingular. Gauss-Jordan Elimination to Solve System of Linear Equations Example: Problem 5, Section 2.6...
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This note was uploaded on 03/21/2012 for the course CS 3333 taught by Professor Boppana during the Fall '11 term at The University of Texas at San Antonio- San Antonio.

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7_Matrices - The right half of the resulting augmented...

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