Rank of a matrix, Gauss-Jordan elimination

[Pages:2]Math 214 ? Spring, 2013

Feb 20

Rank of a matrix, Gauss-Jordan elimination

The Rank of a matrix is the number of nonzero rows in its row echelon form. The Rank Theorem: Let A be the coefficient matrix of a system of linear equations. If the system is consistent, then

number of free variables = n - rank(A).

A matrix is in reduced row echelon form, if: 1. It is in row echelon form 2. The leading entry in each nonzero row is 1. 3. Each column with a leading 1 has zero everywhere else.

The reduced row echelon form of a matrix is unique, and one can reduce any matrix to its reduced row echelon form by elementary row operations. Gauss-Jordan elimination

? Write the augmented matrix of the system of equations ? Use elementary row operations to reduce it to reduced row echelon form ? If the system is consistent, use back substitution to solve the equivalent system that

corresponds to the row-reduced matrix.

Example: Reduce the matrix to its reduced row echelon form

1 3 4 -1 2 -1 5 1

3204

A linear system is called homogeneous, if the constant term in each equation is zero.

A homogeneous system is always consistent, since 0 is always a solution.

Theorem: If [A, 0] is the augmented matrix of a homogeneous system of m linear equations with n variables, where m < n, then the system has infinitely many solutions.

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Exercises: 1. Solve the following system of equations by Gauss-Jordan elimination 2r + s = 3 4r + s = 7 2r + 5s = -1

2. Solve the following system of linear equations by Gauss-Jordan elimination 3w + 3x + y = 1

2w + x + y + z = 1 2w + 3x + y - z = 2

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