Practice Makes Perfect Algebra - Carolyn Wheater [40]
The algebra of matrices allows us to multiply both sides of this equation by the same matrix, just as standard algebra allows us to multiply both sides of an equation by the same number. If then we choose to multiply both sides of this equation by the inverse of the coefficient matrix, we find ourselves with a rapid solution.
The solution of the system of equations is equal to the inverse of the coefficient matrix times the constant matrix.
find our solution by multiplying.
Thus, we know that x = 5 and y = 1.
Although we have used a system of two equations in two variables as our example, the logic of the matrix algebra holds for systems of any size. Since calculators allow us to find inverses and to multiply matrices easily, the technique gives us an easy way to solve systems of any size.
To solve , we first create the equivalent matrix equation.
Matrix algebra then tells us that
Completing the calculation tells us that x = 7, y = −3, and z = 4.
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EXERCISE 13.8
Use inverse matrices to solve each system.
1. −3x + 2y = −20
5x − 3y = 33
2. 2x − 7y = 1
3x + 5y = 17
3. 2x + 3y = 2.9
4x − y = −3.3
4. 2x + 7y = 13
x − 5y = −2
5. 3x − 5y = 36
2x + y = 11
6. 5x + 2y = −21
3x − 4y = −23
7. 2x − y − 3z = 10
x − 2y + 3z = −22
3x + 5y − z = 63
8. x + y − z = −6
x − y + z = 14
x − y − z = 8
9. 2x + 3y = 5
2y + z = 5
x + 3z = −5
10. 5x − 7y + 2z = 40
3x + 2y − z = 5
4x − y + 3z = 27
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Reduced row echelon form
Our final method of solving systems of equations uses a single matrix to contain both the coefficients and the constants that define the system. This matrix is called the augmented matrix. In our earlier example, the matrix of coefficients was and the constant matrix was . To form the augmented matrix, we add the column of constants to the end, or right, of the matrix of coefficients: .
Row operations
While much of the time we wish to perform operations that use an entire matrix as a unit, it is possible to perform some operations which involve only individual rows of a matrix. These operations parallel the algebraic process you used to solve a system of equations. In that process, you will remember, you multiplied through an equation by a constant, added two equations, and when convenient, changed the order of the equations. The corresponding activities for the rows of a matrix are
• Multiply all elements of a row by a constant
• Add the elements of one row to the corresponding elements of another
• Exchange two rows
The last of these, row swapping, is the simplest of the three, and although probably used less often than the other two, it is useful. If we recall that the augmented matrix corresponds to a system of equations, row swapping is equivalent to changing the order in which the equations are written.
If we use the augmented matrix constructed above to illustrate our other row operations, we can multiply all the elements in row 2 by −3. Doing so will transform the matrix into the matrix . We can choose to multiply any row by a constant, and we can choose any constant as the multiplier. What is important for our purposes is to realize that the transformed matrix represents a system of equations with the same solution as our original.
We can add one row of the matrix to another, with the sum replacing one of the rows. For our discussion, we will agree that when we add row X to row Y (where X and Y are row numbers) the sum will replace row Y, that is, we add to a row to change it. The second named row is the row changed, while the first is not altered. If we begin with the matrix and add row 1 to row 2, row 1 will not change, but row 2 will be replaced with the sums of the corresponding elements: .
The row multiplication and row addition operations can be used in tandem, just as multiplication and addition are used in the algebraic solution of systems. If we take our previous row multiplication example and follow it with a row addition, we see these transformations.
If we continue with another row multiplication, we approach