Finding f(A) Using Matrix Capabilities

[themadhatter1]

Homework Statement

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

Homework Equations

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 can't add a constant to a matrix. I'm not sure exactly what I'm supposed to do.

 

[Gear300]

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

 

[themadhatter1]

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

 

[Gear300]

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

 

[themadhatter1]

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

 

[Mark44]

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

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

 

[themadhatter1]

How exactly did you derive 'I's value?

 

[Mark44]

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.

 

[themadhatter1]

Ahh.. ok. I didn't know what an identity matrix was before. But now I know. Interesting.