  # Systems of Linear Equations - Matrix Methods

• A matrix is a rectangular array of numbers , enclosed in brackets . The numbers are called the
entries of the matrix. Entries are identified by their row and column position. Rows run
horizontally, columns run vertically.

• Examples : • An augmented matrix can be used to represent a system of equations.
The system is represented as Notice that equation 1 becomes row 1, equation 2 becomes row 2, the x terms are in column 1, the
y terms are in column 2, and the equal signs are represented by the vertical line .

• Write the augmented matrix that represents the following system of equations. • Write the system of equations that corresponds to the following augmented matrix. • Write the system of equations that corresponds to the following augmented matrix. • To solve a system of equations using its augmented matrix representation, we will transform the
original augmented matrix into a form similar to the previous example . This will allow us to read
the solutions of the system .

• There are three row operations that can be applied to an augmented matrix. These correspond to
algebraic operations that can be applied to the corresponding system of equations.

• Row operations
1. Interchange any two rows .
2. Replace any row by a nonzero constant multiple of that row .
3. Replace any row by the sum of that row and a constant multiple of another row.

• Matrix method - an example.
Solve Step 1 : Write the augmented matrix. Step 2 : Use row operations to transform the augmented matrix into the form which has solutions x = a, y = b . The last augmented matrix corresponds to the system which has solution x = 1, y = 2.

Check :
5(1) + 10(2) = 25
10(1) + 12(2) = 34

• The strategy for transforming the original augmented matrix using row operations:

1. Place a 1 in row1, column 1
2. Place 0's in all other entries in column 1 - leaving the 1 in row 1, column 1 unchanged
3. Place a 1 in row 2, column 2
4. Place 0's in all other entries in column 2 - leaving the 1 in row 2, column 2 unchanged
5. Continue this pattern. Place a 1 in row n, column n. Place 0's in all other entries of column n -
leaving the 1 in row n, column n unchanged.
6. If a row is obtained that contains only 0's to the left of the vertical bar, place it at the bottom of
the matrix.

A matrix generated using the strategy outlined above is said to be in row-echelon form.

• Solve by writing the augmented matrix in row-echelon form.  The solution is x = 1, y = 0, z = 2 .

• Example of an inconsistent system.
Solve  Notice that row 2 corresponds to the equation 0 = 31, a contradiction. Therefore, this
system has no solution. It is inconsistent.

• Example of a consistent system with dependent equations. Notice that row 2 corresponds to the equation 0 = 0, an identity. This indicates that
equation 2 can be derived from equation 1. They are equivalent equations .

Any point on the line is a solution of the system.

Solutions: where y is any real number . ( y is called a parameter.)

Give three different solutions for this system.

• Use the matrix method to solve the following system. Prev Next

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