Tutorial: A Module for Linear Regression

Example: Solving a System of Linear Equations

Because the syntax of the SAS/IML language is similar to the notation used in linear algebra, it is often possible to directly translate mathematical methods from matrix-algebraic expressions into executable SAS/IML statements. For example, consider the problem of solving three simultaneous equations:

StartLayout 1st Row 1st Column 3 x 1 minus x 2 plus 2 x 3 2nd Column equals 3rd Column 8 2nd Row 1st Column 2 x 1 minus 2 x 2 plus 3 x 3 2nd Column equals 3rd Column 2 3rd Row 1st Column 4 x 1 plus x 2 minus 4 x 3 2nd Column equals 3rd Column 9 EndLayout

These equations can be written in matrix form as

Start 3 By 3 Matrix 1st Row 1st Column 3 2nd Column negative 1 3rd Column 2 2nd Row 1st Column 2 2nd Column negative 2 3rd Column 3 3rd Row 1st Column 4 2nd Column 1 3rd Column negative 4 EndMatrix Start 3 By 1 Matrix 1st Row x 1 2nd Row x 2 3rd Row x 3 EndMatrix equals Start 3 By 1 Matrix 1st Row 8 2nd Row 2 3rd Row 9 EndMatrix

and can be expressed symbolically as

where is the matrix of coefficients for the linear system. Because is nonsingular, the system has a solution given by

bold x equals bold upper A Superscript negative 1 Baseline bold c

This example solves this linear system of equations.

Define the matrices and . Both of these matrices are input as matrix literals; that is, you type the row and column values as discussed in Chapter 2, Understanding the SAS/IML Language.
```
proc iml;
a = {3  -1  2,
     2  -2  3,
     4   1 -4};
c = {8, 2, 9};
```
Solve the equation by using the built-in INV function and the matrix multiplication operator. The INV function returns the inverse of a square matrix and * is the operator for matrix multiplication. Consequently, the solution is computed as follows:
```
x = inv(a) * c;
print x;
```
Figure 1: The Solution of a Linear System of Equations

x

3
5
2
Equivalently, you can solve the linear system by using the more efficient SOLVE function, as shown in the following statement:
```
x = solve(a, c);
```

x
3
5
2

After SAS/IML executes the statements, the rows of the vector x contain the , and values that solve the linear system.

You can end PROC IML by using the QUIT statement:

quit;

Last updated: May 07, 2026