10.3 Systems of Linear Equations: Matrices

A matrix is defined as a rectangular array of numbers, Column 1Column 2 Column jColumn n Row 1 Row 2 Row 3 Row 4

Augmented Matrix:

Row Operations on an Augmented Matrix 1. Interchange any two rows. 2. Replace a row by a nonzero multiple of that row. 3. Replace a row by the sum of that row and a constant multiple of some other row.

Echelon Form of an Augmented Matrix

Solve Find the echelon form. Find the augmented matrix: R 2 =-2 R 1 + R 2

R 2 =R 2 /3 R 3 =-4R 2 +R 3

R 3 =R 3 *(-3/25) The third row of the matrix represents the equation z =-7/25. Substituting this into the equation represented by the second row we get:

Let z =-7/25, y =-44/25 in the first: Solution is:

Solve using a graphing utility.

Substitute z = 5 into the second. Substitute z=5, y =-2 into the first.

Solution is (x, y, z) = (1, -2, 5).

Dependent system: Infinitely many solutions. Solve using a graphing utility: Using rref(.) function we get:

Solve for y from the second: Solve for x from the first: Solution is (x, y, z) = (18/5 - (7/5)z, 7/5 + (2/5)z, z)