# Thread: Plugging Matrixes into Functions

1. ## Plugging Matrixes into Functions

I'm taking linear algebra and I'm not really understanding how putting a matrix into a function works. We learned that f(x) = A^2 + 2A + 1 means square matrix A + 2* matrix A + Identity Matrix. Our teacher said you can basically plug anything into a function. But does the function still obey the laws of algebra?

For example can you factor a function? I have a question that says show p1(A) = p2(A)p3(A) for any square matrix A. It tells me p1(x) = x^2 + 9 and p2(x) = x + 3 and p3(x) = x - 3. I verified it for a specific 2x2 matrix A, but am not sure how to generalize. I would like to generalize it even farther than they are asking to factor anything like I can with regular algebra.

I have a similar question later, show that a square matrix A satisfies A^2 - 3A + I = 0 then A^-1 = 3I - A. It is trivial to show it works for a specific matrix A, and I would like to understand how they came up with the 2nd equation, and not merely that it is true, so I can understand what operations are valid.

I don't really know the name of what I'm having trouble with, so I wasn't really able to find anything useful on the internet. I'm taking it at a community college, so the people in the math lab can't really help me, in fact I work in our math lab myself.
Thanks so much for your help!

2. Originally Posted by jjoshua2
I'm taking linear algebra and I'm not really understanding how putting a matrix into a function works. We learned that f(x) = A^2 + 2A + 1 means square matrix A + 2* matrix A + Identity Matrix. Our teacher said you can basically plug anything into a function. But does the function still obey the laws of algebra?
Well, you can plug anything you can add and multiply and multiply by numbers into a polynomial. You can even put them into more complicated functions if you can write the functions as infinite power series.

For example can you factor a function? I have a question that says show p1(A) = p2(A)p3(A) for any square matrix A. It tells me p1(x) = x^2 + 9 and p2(x) = x + 3 and p3(x) = x - 3.
I hope it didn't tell you that! $(x+3)(x-3)= x^2- 9$, not $x^2+ 9$! Was that what you meant?

I verified it for a specific 2x2 matrix A, but am not sure how to generalize. I would like to generalize it even farther than they are asking to factor anything like I can with regular algebra.
Just be careful about "commutativity". Numbers commute under multiplication (ab= ba) while matrices do not (in general $AB\ne BA$).
$(A+B)(A+B)= A^2+ AB+ BA+ B^2$ but that is NOT $A^2+ 2AB+ B^2$ as it would be for numbers.
Matrices are just like numbers except that multiplication is not commutative and not every matrix has a multiplicative inverse. For example if you know AB= 0, for matrices A, B, and the zero matrix, you cannot conclude that A= 0 or B= 0.

I have a similar question later, show that a square matrix A satisfies A^2 - 3A + I = 0 then A^-1 = 3I - A. It is trivial to show it works for a specific matrix A, and I would like to understand how they came up with the 2nd equation, and not merely that it is true, so I can understand what operations are valid.
A(A-3)= (A-3)A= -I so that A(3-A)= (3-A)A= I. By definition, $A^{-1}$ is the vector such that $A^{-1}A= AA^{-1}= I$.

I don't really know the name of what I'm having trouble with, so I wasn't really able to find anything useful on the internet. I'm taking it at a community college, so the people in the math lab can't really help me, in fact I work in our math lab myself.
Thanks so much for your help!