7.3 & 7.4 – MATRICES AND SYSTEMS OF EQUATIONS

I N THIS SECTION, YOU WILL LEARN TO  Write a matrix and identify its order  Perform elementary row operations on matrices  Use matrices and Gaussian elimination to solve systems of linear equations  Use matrices and Gauss-Jordan elimination to solve systems of linear equations

DEFINITION OF A MATRIX: Model Fittings: Prepare field trip slips/medical release forms for each fitting scheduled during

DEFINITION OF A MATRIX:

ORDER OF THE MATRIX: 1) 2) 3) 4)

FORMS OF A MATRIX: 1) A matrix is derived from a system of linear equations which can be represented in a coefficient matrix or an augmented matrix form:

FORMS OF A MATRIX: a) System of linear equations: b) Coefficient Form: c) Augmented Form:

ELEMENTARY ROW OPERATIONS: 1) These operations are used to solve for the values in the system of linear equations. a) Interchange two rows b) Multiply a row by a nonzero constant c) Add a multiple of a row to another row

MATRIX EXAMPLES: Solve: Linear MethodMatrix Method * We are going to solve this problem using the linear system method and the matrix elementary row operation method side by side.

S TEP 1: Add the first row to the second row: LINEAR METHODMATRIX METHOD Add the first equation to the second equation:

S TEP 2: Add times the first row to the third row: Add times the first equation to the third equation: LINEAR METHODMATRIX METHOD

S TEP 3: Add the second row to the third row: Add the second equation to the third equation: MATRIX METHODLINEAR METHOD

S TEP 4: Multiply the third row by : Multiply the third equation by : MATRIX METHODLINEAR METHOD

STEP 5:  Use back substitution to solve for the remaining variables.

ROW ECHELON AND REDUCED ROW ECHELON FORMS:  A matrix in row-echelon form has the following properties: a) All rows entirely of zeros is at the bottom of the matrix. b) For any rows not made entirely of zeros, the first nonzero entry is 1. (leading 1) c) For two successive rows, the leading 1 in the higher row is farther to the left than the lower row.

ROW ECHELON AND REDUCED ROW ECHELON FORMS:  Row Echelon Form:  Not in Row Echelon Form:

ROW ECHELON AND REDUCED ROW ECHELON FORMS:  Reduced Row Echelon Form : a matrix is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1.  Not in Row Echelon Form:

GAUSSIAN ELIMINATION WITH BACK SUBSTITUTION: ( NAMED AFTER CARL FRIEDRICH GAUSS) 1) Write the augmented matrix of the system of linear equations. 2) Use elementary row operations to rewrite the augmented matrix in row- echelon form. 3) Write the system of linear equations corresponding to the matrix in row- echelon form, and use back-substitution to find the solution.

EXAMPLE #1: Solve:

EXAMPLE #1:

GAUSS-JORDAN ELIMINATION:  A second method of elimination is named after Carl Friedrich Gauss and Wilhelm Jordan. You continue with the reduction process until a reduced row- echelon form is obtained. Solve using this method.

GAUSS-JORDAN ELIMINATION EXAMPLE:

SYSTEM WITH NO SOLUTION:  It is possible for a system of linear equations to have no solution. If you obtain a row with zeros except for the last entry, it is unnecessary to continue the elimination process. The system has no solution, or is inconsistent. Example:

SYSTEM WITH INFINITE MANY SOLUTION:  It is possible for a system of linear equations to have an infinite number of solutions. Example:

SYSTEM WITH INFINITE MANY SOLUTION: Example #1:

SYSTEM WITH INFINITE MANY SOLUTION: