Matrix Arithmetic

Matrix Arithmetic - Matrix Arithmetic Operations One of the...

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One of the biggest impediments that some people have in learning about matrices for the first time is trying to take everything that they know about arithmetic of real numbers and translate that over to matrices. As you will eventually see much of what you know about arithmetic of real numbers will also be true here, but there is also a few ideas/facts that will no longer hold here. To make matters worse there are some rules of arithmetic of real numbers that will work occasionally with matrices but won’t work in general. So, keep this in mind as you go through the next couple of sections and don’t be too surprised when something doesn’t quite work out as you expect it to. This section is devoted mostly to developing the arithmetic of matrices as well as introducing a couple of operations on matrices that don’t really have an equivalent operation in real numbers. We will see some of the differences between arithmetic of real numbers and matrices mentioned above in this section. We will also see more of them in the next section when we delve into the properties of matrix arithmetic in more detail. Okay, let’s start off matrix arithmetic by defining just what we mean when we say that two matrices are equal. Definition 1 If A and B are both matrices then we say that A = B provided corresponding entries from each matrix are equal. Or in other words, A = B provided for all i and j . Matrices of different sizes cannot be equal. Example 1 Consider the following matrices. For these matrices we have that and since they are different sizes and so can’t be equal. The fact that C is essentially the first column of both A and B is not important to determining equality in this case. The size of the two matrices is the first thing we should look at in determining equality.
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Next, A = B provided we have . If then we will have . Next we need to move on to addition and subtraction of two matrices. Definition 2 If A and B are both matrices then is a new matrix that is found by adding/subtracting corresponding entries from each matrix. Or in other words, Matrices of different sizes cannot be added or subtracted. Example 2 For the following matrices perform the indicated operation, if possible. (a) (b) (c) Solution (a) Both A and B are the same size and so we know the addition can be done in this case. Once we know the addition can be done there really isn’t all that much to do here other than to just add the corresponding entries here to get the results. . (b) Again, since A and B are the same size we can do the difference and as like the previous part
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there really isn’t all that much to do. All that we need to be careful with is the order. Just like with real number arithmetic is different from . So, in this case we’ll subtract the entries of A from the entries of B . (c)
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This note was uploaded on 11/06/2011 for the course MATH 211 taught by Professor Staff during the Spring '11 term at Rutgers.

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Matrix Arithmetic - Matrix Arithmetic Operations One of the...

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