2.4.1 “Easy” Systems of Linear Equations

Before considering the solution of general systems of linear equations, we will begin with a discussion of some problems that have special characteristics that make them particularly easy to solve.

Diagonal Systems.

Consider the following system.

(2.8)

It should be clear what the values of x, y, and z must be. You can literally read the solution from the system of equations.

The coefficient matrix for this problem is

and is a diagonal matrix; each nondiagonal element has the value zero. A general n x n diagonal matrix has the form

If this diagonal matrix were the coefficient matrix of an n x n linear system, then we could easily find a solution by solving the first equation for the first variable, the second equation for the second variable, and so on.

Upper Triangular Systems.

Now consider this system:

(2.9)

The first equation involves all of the variables, but if we scan down the list, we find that the last equation involves only z. It is easy to determine the value of the variable z using the third equation. If we substitute that value for z in the second equation, we obtain an equation involving only y. Once we have values for y and z, we can use the first equation to solve for x.

The coefficient matrix for this system is

This coefficient matrix is an upper triangular matrix; all of the elements below the diagonal have value zero.

The general form for an upper triangular matrix is

The elements in the first row can have any value, and each row begins with one more required zero than the row above it. By the time we get to the final row, only a single nonzero entry is allowed. If the coefficient matrix of an n x n linear system is an upper triangular matrix, then we can easily find a solution by solving the last equation in the system for the n^th unknown, substituting this value into the equation directly above it (thereby causing the n - 1^st equation to have only one unknown), and continuing in this manner from bottom to top. This process of solving equations from bottom to top is called back substitution.

Send comments on material to Cynthia Lanius

These pages are maintained by Hilena Vargas (hvargas@rice.edu)
Updated: February 23, 2001