Matrices and systems of Equations

1. Matrices and systems of Equations

2. Definition of a Matrix • * Rectangular array of real numbers • m rows by n columns • * Named using capital letters • * First subscript is row, second subscript is column

3. Terminology • * A matrix with m rows and n columns is called a matrix of order m x n. • A square matrix is a matrix with an equal number of rows and columns. Since the number of rows and columns are the same, it is said to have order n. • * The main diagonal of a square matrix are the elements from the upper left to the lower right of the matrix. • * A row matrix is a matrix that has only one row. • * A column matrix is a matrix that has only one column. • * A matrix with only one row or one column is called a vector.

4. x y z rhs 1 1 -1 1 3 -2 1 3 4 1 -2 9 [ ] Converting Systems of Linear Equations to Matrices Each equation in the system becomes a row. Each variable in the system becomes a column. The variables are dropped and the coefficients are placed into a matrix x + y - z = 13 x - 2y + z = 34 x + y - 2z = 9

5. x y z 1 1 -1 3 -2 1 4 1 -2 x y z rhs 1 1 -1 1 3 -2 1 3 4 1 -2 9 ] ] [ [ Augmented Matrix: standard form of a matrix derived from a system of linear equations Coefficient Matrix: standard form of a matrix not including the constant terms

6. Convert the following into augmented matrix 4x +2y - z = 44 x - y + 3z = 9 x + 3y - z = 0 -x + 2y - 6z = 1 x - y + 3z = 1 x + y - z = 4 • x - z = 13 • 2y + z = 34 • x + y = 9 z = 1 -2y = 24 x = 6

7. { [ ] x-2y+3z=9 y+3z=5 z=2 1 - 2 3 9 0 1 3 5 0 0 1 2 Back-Substitution x - 2 (-1) + 3 (2) = 9 x - 2(-1) + 3(2) = 9 x + 2 + 6=9 x=1 y + 3(2) = 5 y = -1 x=1, y=-1, z=2

8. Practice:Use back substitution to find the value of x, y, z 1 + 2 + 3 4 0+ 1 + 4 6 0 + 0+ 1 2 1 + 3 + 1 -6 0 + 1 - 9 12 0 + 0+ 1 3 X=2, y=-2, z=2 X=-126, y=39, z=3 1 - 0 + 3 1 0+ 1 + 5 3 0 + 0+ 1 -7 1 - 4 + 1 4 0+ 1 - 3 2 0 + 0+ 1 7 X=22, y=38, z=-7 X=89, y=23, z=7