Complete for Example 4.15. Find numbers a for which the following homogeneous system has a nontrivial solution and solve the system for these values of The determinant of the coefficient matrix is...

Complete

for

Example 4.15. Find numbers a for which the following homogeneous system has a nontrivial solution and solve the system for these values of

The determinant of the coefficient matrix is

after expanding by minors using the first column. The value of the determinant is then

So


or

and these values of a are the only values for which the given homogeneous system has a nontrivial solution. If

reduction of the coefficient matrix to upper triangular form gives

and so the complete solution is

with

arbitrary, and the null space is spanned by

The case of

is left to the exercises.

If the Symbolic Toolbox is available in your MATLAB distribution, there are many possibilities for performing symbolic manipulation with linear algebra. The following is a partial quotation from the help system for the toolbox.