# Matrix Multiplication

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- First, understand when two matrices can and cannot be multiplied together
- Learn the pattern-based approach for systematically multiplying two matrices
- Understand why the associative property of multiplication FAILS for matrices, and thus multiplication order matters
- For square matrices, learn how the identity matrix and the zero matrix act just like the numbers 1 and 0 do for number multiplication, respectively

Multiplying matrices is a bit tedious, but its a staple item on the menu of things your teacher will serve up on a test, when you study matrices. It's also occasionally useful to know in the future, having a few important applications in probability and statistics. This lesson covers the basics and then some, for everything there is to know about multiplying two matrices together.

Practice problems and worksheet coming soon!

## Multiplication Meets Misery

Multiplying matrices will give you new appreciation for multiplying good ol' integers together. Regrettably, they are quite a bit more complex to multiply than numbers, for a few reasons.First, it is difficult to picture or interpret what matrix multiplication means, unlike integers. For example, when we were in grammar school and we started memorizing facts like $4 \times 3$, we could at least interpret the result of $12$ by thinking about having $4$ sets of $3$ or vice versa. Visually, we might interpret $4 \times 3$ as something like this:In any case, multiplying two numbers had a physical interpretation to accompany the rote memorization facts that we were asked to know. With matrix multiplication, while there is reason and purpose to doing it, it lacks the intrinsic interpretation that integer multiplication has.Second, order matters. That's right - it sounds incredulous, after spending so long with numbers and the well-known and convenient Commutative Property of Multiplication, but it's true. $5 \times 6$ may equal $6 \times 5$, but matrix $A$ times matrix $B$ is NOT equal to matrix $B$ times matrix $A$ (technically this CAN happen but it is unlikely and uncommon).Third, if we have to do it by hand, there are many little calculations involved, and we need to make zero mistakes. As we learn the process, we'll see why each number in the resulting matrix is calculated using several numbers from each of the two parent matrices, which is why it's easy to slip. Of course, if you follow my method and do a little practice, you won't fall victim to this inherent pitfall.## Before We Multiply

A very important prerequisite for multiplying matrices is ensuring that you can even multiply them in the first place. Unlike integers, you cannot just pick any two and multiply them together. Matrices actually have to be compatible before they can be multiplied.Recall that each matrix has what we called a dimension, given by the number of rows and columns the matrix has. Specifically, a matrix with $m$ rows and $n$ columns is said to have dimension $m$ by $n$.Two matrices can only be multiplied if they have a match on certain dimension numbers.## Matrix Multiplication How-To

Matrix multiplication is a matter of computing each entry in the resulting matrix one-at-a-time. Because this is the case, it will quickly become clear why it is beneficial for us to map out and know what the dimensions of the resulting matrix will be using the Bracket Method, before we even multiply.Each entry in the resulting matrix is obtained by taking the dot product of the first matrix's row with the second matrix's column. In fact, this is exactly the reason why these dimensions must match! If you aren't familiar with "dot" products, it means to take the two sets of numbers and multiply each corresponding number from each set together, and add up all the results. An example or two will make this process clearer.In this illustrative example, we will multiply a $2 \times 4$ matrix with a $4 \times 2$ matrix, the product of which yields a $2 \times 2$ result.## Properties of Matrix Multiplication

There are three major properties of matrix multiplication that we need to understand, regardless of whether or not our teacher will let us use the aide of a calculator to get it done.Commutative Property FailureWe are not strangers to the fact that $3 \times 7$ and $7 \times 3$ are both equal to $21$. In early Algebra, we formally defined this multiplication property for all numbers and variables, and we called it the Commutative Property of Multiplication. Formally, we say that $a \cdot b$ is equal to $b \cdot a$ for any two real numbers $a$ and $b$. Oddly enough, this property fails for matrix multiplication. While it is possible to find two matrices $A$ and $B$ such that $AB = BA$, it is not guaranteed to be true, and indeed well over 90% of the matrix multiplication you do will be with matrices such that $AB$ is not equal to $BA$.- $AB$ will never equal $BA$ unless $A$ and $B$ are square matrices of the same dimension
- Even if $A$ and $B$ are square matrices of the same dimension, $AB$ and $BA$ will only be the same in very specific circumstances. One such common circumstance is that $A$ and $B$ are inverses of one another. The other ones are more coincidental and not worth outlining here and now. The takeaway here is that there is no guarantee that $AB$ and $BA$ will give the same result.

## Put It To The Test

Here are a few examples on how you can expect this material to show up on a quiz or test. We'll do three straight-up multiplication exercises followed by some concept-based questions.Examples 6-8Multiply each given matrix product.

- Understand specifically when two matrices can and can't be multiplied together
- Use the method of dot products to obtain the answer for the product of two matrices, one entry at a time
- Be lazy and use a calculator, if you can - some teachers don't mind, and even if you have to do it long-hand, it's nice to check your work
- Be familiar with the three properties of matrix multiplication - that order matters (Commutative Property fails), and products involving the zero or identity matrices

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