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Systems of Linear Equations - Matrix Methods

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 re present 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 fol lowing 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. Inter change any two rows .
2. Replace any row by a non zero 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.

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