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Finding the characteristic polynomial of a matrix

When we need to find eigenvalues and eigenvectors in algebra, we start by finding the characteristic polynomial of a matrix. In this problem, we will find the characteristic polynomial of a matrix, A.

A = [(2, 0), (0, -1)]

We will also discuss what is the characteristic polynomial, the equation that describes it.

Solution Preview

When we study square matrices in linear algebra, we are studying a grid of numbers. However, we can also associate a polynomial with each matrix that describes certain qualities of the matrix. This is called the characteristic polynomial. The characteristic polynomial of a matrix can tell us about its eigenvalues, its determinate and its trace. So you can see, to be able to find this polynomial is very important to our study of linear algebra.

So let's start with a simple matrix. [(2, 0), (0, -1)]. For clarification, the numbers are grouped in column vectors. The most important equation for finding our characteristic polynomial is det[(lambda)I - A]. For a quick review, what is I?
I is the identity matrix, [(1, 0), (0, 1)]. (lambda)I ...

Solution Summary

In this discussion, the process for finding the characteristic polynomial of a matrix was described. Using a simple matrix, a solution was found using the equation
det[(lambda)I - A], yielding a surprising result of eigenvalues as well. This process was described in a simple straightforward manner geared to those studying undergraduate linear algebra for engineering and many other applied fields, as well as those planning to continue on in mathematics.