Chapter 8: Lesson 8.1 Matrices & Systems of Equations The size or “order” of a matrix is determined by the number of rows and columns. The matrix above is a (2 x 3) matrix.

4x – 3y = -5 -x + 3y = 12 This represents a coefficient matrix for the system of equations. This represents an augmented matrix for the system of equations.

Elementary Matrix Row Operations It is acceptable to interchange any 2 rows. It is acceptable to multiply any row by a NONZERO constant. It is acceptable to add a multiple of a row to any other row.

Gaussian Elimination Make all the main diagonal entries = 1 by using elementary row operations. Make all the entries below the main diagonal entries = 0 by using elementary row operations. Process Start with the first column and make the first main diagonal entry = 1 by using elementary row operations. Make all entries below that main diagonal entry = 0 by using elementary row operations. Repeat steps 1 and 2 for any remaining columns from left to right.

Row Echelon Form of a Matrix Any rows consisting entirely of zeros occurs at the bottom of the matrix. Each row that does not consist entirely of zeros has the first non zero entry as 1 (leading 1). For 2 successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. A matrix in row echelon form is in reduced row echelon form if every column that has a leading 1 has zeros in every position above and below its leading zero.

#68 Solve the system below by using Gaussian Elimination -x + y = 4 2x – 4y = -34

See example 7 on page 544 to see an example of what NO SOLUTION looks like.