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Dimensions of a Product Matrix

Matrix multiplication is a necessary part of mathematics and even if it does not seem so now, it can be very fun!

But to get to the fun portion, you must learn and master the basics. So, without further ado, let us discuss the dimensions of a product of two matrices.

We denote matrix dimensions as row by column. So a matrix denoted $(3 \times 2)$ would look like this:

Now when we multiply matrices together there are certain rules that we must adhere to. Namely, that matrices that can be multiplied must follow the form: $(m \times n) \times (n \times p)$

This rule also necessarily implies that matrix multiplication is not commutative

Let's explore what I've just stated a little deeper!

Matrix Multiplication when $A \times B=(m \times n) \times (n \times p)$

You have Matrix A and B, respectively:

We will have to use the dot product to solve this expression, so if you need to review matrix multiplication using dot product then feel free to click the link before moving on.

We have a 2x3 and a 3x2, which satisfies our $(m \times n) \times (n \times p)$ so now we can move on to finding the resultant matrix.

The first row of Matrix A dot the first column of matrix B $= a_1$

The first row of Matrix A dot the second column of Matrix B $= a_2$

The second row of Matrix A dot the first column of Matrix B $= a_3$

The second row of Matrix A dot the second column of matrix B $= a_4$

Thus, the final matrix will be of the form $(2\times 2)$ and will be made in the following lay:

You might notice something curious about this final matrix, namely about its dimensions. It's a $(2 \times 2)$ which was the $m$ portion of matrix A and the $p$ portion of matrix B, according to our rule. If you think this is will consistently be the outcome of proper matrix multiplication, then you are correct!

The final matrix will always result in dimensions of $(m \times p)$

Now, let's look at why matrix multiplication isn't commutative.

Matrix Multiplication Non-commutative

This concept actually is quite simple to understand! Taking the example from above we can easily see that if

$A = (2 \times 3)$ and $B = (3 \times 2)$ then $(A \times B)$ will produce a $(2 \times 2)$ matrix because it follows our $(m \times n) \times (n \times p)$ rule, but $(B \times A)$ does not. Simply because A's row dimension doesn't match B's column dimension.

Try it out on your own and see for yourself!

(Video) Determine the Dimensions of Each Product Matrix

by James Elliott

This video by James Elliott walks you through a couple of examples of finding the dimensions of a product matrix.

Summary

Given the following matrices, we want to find the product of: $AB$, $CA$, $DB$, and $BD$.

$A=\begin{bmatrix} 2&-2&3 \\ -5&1&-4 \\ \end{bmatrix}$

$B=\begin{bmatrix} 4&0 \\ -1&5 \\ 3&-1 \\ \end{bmatrix}$

$C=\begin{bmatrix} -1&0&3 \\ \end{bmatrix}$

$D=\begin{bmatrix} -2 \\ 0 \\ \end{bmatrix}$

First, we need to establish the dimensions of these matrices. Matrix $A$ is a $2x3$ matrix. Matrix $B$ is a $3x2$ matrix. Matrix $C$ is a row matrix, so its dimensions are $1x3$. Matrix $D$ is a column matrix with dimensions $2x1$. We need to figure out the dimensions for each of the above product matrices.

For the product matrix $AB$, we will be taking Matrix $A$, a $2x3$ matrix, and multiplying it by Matrix $B$, a $3x2$ matrix.

$$2x3 \cdot 3x2$$

First, we need to establish if we can find the product of these two matrices. Because the inner two values are the same, $3$, we can find the product of Matrix $A$ and Matrix $B$. The dimensions of the product matrix will be the two outer values: $2$ rows and $2$ columns. So, the dimensions of product matrix $AB$ are $2x2$.

The next product matrix we are looking at are for product matrix $CA$. The dimensions of Matrix $C$ are $1x3$, and the dimensions of Matrix $A$ are $2x3$.

$$1x3 \cdot 2x3$$

Since the middle values here are not the same, $3$ and $2$, we cannot find the product of Matrix $C$ and Matrix $A$.

Next up, we are looking at the product matrix $DB$. Matrix $D$ has dimensions of $2x1$, and Matrix $B$ has dimensions of $3x2$.

$$2x1 \cdot 3x2$$

Since the inner values are not the same, we cannot find the product of these two matrices.

The last example looks at the product matrix $BD$. Remember that matrix multiplication is not commutative so order matters!

$$3x2 \cdot 2x1$$

The two inner values, $2$, are the same, so we know we can find the product of these two matrices, in this particular order. The product matrix will have the dimensions of the two outer values, $3$ and $1$. So, the product matrix, $BD$, has the dimensions of $3x1$.

Now that we've learned the basics of the row and column matrix, we can move forward into more complicated types of matrix multiplication.

Dimensions of a Product Matrix

There are three simple steps to follow when trying to figure out the dimensions of a product matrix.

#1 The first step is to establish the dimensions of the matrices.

$$\begin{bmatrix} 2&3 \\ 1&4 \\ \end{bmatrix} \times \begin{bmatrix} 4&6&8 \\ 9&3&4 \\ \end{bmatrix}$$

The dimensions of these matrices are $2x2$ and $2x3$, respectively. Look at the number of rows and the number of columns.

#2 The second step is to make sure that the center values are the same. If the center values are not the same, the two matrices cannot be multiplied. These two values are the columns in the first matrix and the rows in the second matrix.

$$2x2 \times 2x3$$

The two center values are the same, so these two matrices can be multiplied.

#3 The third step tells you that the outer values are the dimensions for the product matrix.

$$2x3$$

This is the resulting matrix.