# Matrix question

1. Jul 6, 2010

1. The problem statement, all variables and given/known data
Use the matrix capabilities of a graphing utility to find:
$$f(A)=a_{0}I_{n}+a_{1}A+a_{2}A^2+\cdots+a_{n}A^n$$

1.
$$f(x)=x^2-5x+2$$
$$A=\left[\begin{array}{cc}2&0\\4&5\end{array}\right]$$

2. Relevant equations

3. The attempt at a solution

Well, I know the answer is
$$\left[\begin{array}{cc}-4&0\\8&2\end{array}\right]$$
However, I don't know how to get it.

I would think you would do A^2-5A+2 however you cant add a constant to a matrix. I'm not sure exactly what I'm supposed to do.

2. Jul 6, 2010

### Gear300

The constant 2 might be
$$\left[\begin{array}{cc}2&2\\2&2\end{array}\right]$$

3. Jul 6, 2010

Nope, thats not it. Just tried it and it's wrong, not sure how it would be it though.

4. Jul 6, 2010

### Gear300

A constant by itself might signify that it is in operation with an identity. So 2 might be
$$\left[\begin{array}{cc}2&0\\0&2\end{array}\right]$$

5. Jul 6, 2010

oh, ok. That's turns out to be right. Thanks!

6. Jul 6, 2010

### Staff: Mentor

Right. The polynomial is f(A) = A2 - 5A + 2I.

7. Jul 6, 2010

How exactly did you derive 'I's value?

8. Jul 6, 2010

### Staff: Mentor

I is the 2 x 2 identity matrix, defined as
$$\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$$

Since A is given as a 2 x 2 matrix, the appropriate identity matrix must also be 2 x 2. If A were given as a 3 x 3 matrix, you would need to use the 3 x 3 identity matrix, which is defined as
$$\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$$

The form of the identity matrix to use depends on the size of the square matrices being used in the problem.

9. Jul 6, 2010