10.3 Systems of Linear Equations:

Matrices

A matrix is defined as a rectangular array of numbers,

Column 1 Column 2 Column j Column 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:

R2=-2 R1+ R2

R2 =R2/3

R3 =-4R2+R3

R3=R3*(-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)