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### Course: Algebra (all content)>Unit 20

Lesson 9: Properties of matrix multiplication

# Matrix multiplication dimensions

Learn about the conditions for matrix multiplication to be defined, and about the dimensions of the product of two matrices.

#### What you should be familiar with before taking this lesson

A matrix is a rectangular arrangement of numbers into rows and columns. Each number in a matrix is referred to as a matrix element or entry.
The dimensions of a matrix give the number of rows and columns of the matrix in that order. Since matrix $A$ has $2$ rows and $3$ columns, it is called a $2×3$ matrix.
If this is new to you, we recommend that you check out our intro to matrices.
In matrix multiplication, each entry in the product matrix is the dot product of a row in the first matrix and a column in the second matrix.
$\begin{array}{cccccc}& & & & \stackrel{\to }{{b}_{1}}& \stackrel{\to }{{b}_{2}}\\ & & & & ↓& ↓\\ \\ \begin{array}{c}\stackrel{\to }{{a}_{1}}\to \\ \stackrel{\to }{{a}_{2}}\to \end{array}& \left[\begin{array}{c}1\\ 2\end{array}& \begin{array}{c}7\\ 4\end{array}\right]& \cdot & \left[\begin{array}{c}3\\ 5\end{array}& \begin{array}{c}3\\ 2\end{array}\right]& =& \left[\begin{array}{c}\stackrel{\to }{{a}_{1}}\cdot \stackrel{\to }{{b}_{1}}\\ \stackrel{\to }{{a}_{2}}\cdot \stackrel{\to }{{b}_{1}}\end{array}& \begin{array}{c}\stackrel{\to }{{a}_{1}}\cdot \stackrel{\to }{{b}_{2}}\\ \stackrel{\to }{{a}_{2}}\cdot \stackrel{\to }{{b}_{2}}\end{array}\right]\\ \\ & A& & & B& & & C\end{array}$
If this is new to you, we recommend that you check out our matrix multiplication article.

#### What you will learn in this lesson

We will investigate the relationship between the dimensions of two matrices and the dimensions of their product. Specifically, we will see that the dimensions of the matrices must meet a certain condition for the multiplication to be defined.

## When is matrix multiplication defined?

In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
To see why this is the case, consider the following two matrices:
$A=\left[\begin{array}{rr}1& 3\\ 2& 4\\ 2& 5\end{array}\right]$ and $B=\left[\begin{array}{rrrr}1& 3& 2& 2\\ 2& 4& 5& 1\end{array}\right]$
To find $AB$, we take the dot product of a row in $A$ and a column in $B$. This means that the number of entries in each row of $A$ must be the same as the number of entries in each column of $B$.
$A=\left[\begin{array}{rr}1& 3\\ 2& 4\\ 2& 5\end{array}\right]$ and $B=\left[\begin{array}{rrrr}1& 3& 2& 2\\ 2& 4& 5& 1\end{array}\right]$
Note that if a matrix has two entries in each row, then the matrix has two columns. Similarly, if a matrix has two entries in each column, then it must have two rows.
So, it follows that in order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

1) $A=\left[\begin{array}{rr}2& 4\\ 6& 4\\ 7& 3\end{array}\right]$ and $B=\left[\begin{array}{rr}2& 1\\ 8& 5\end{array}\right]$.
Is $AB$ defined?

2) $C=\left[\begin{array}{rrrr}5& 3& 6& 1\\ 6& 8& 5& 3\end{array}\right]$ and $D=\left[\begin{array}{rrr}2& 1& 8\\ 7& 5& 5\end{array}\right]$.
Is $CD$ defined?

3) $A$ is a $4×2$ matrix and $B$ is a $2×3$ matrix.
Is $AB$ defined?

Is $BA$ defined?

## Dimension property

The product of an $m×n$ matrix and an $n×k$ matrix is an $m×k$ matrix.
Let's consider the product $AB$, where $A=\left[\begin{array}{rr}1& 3\\ 2& 4\\ 2& 5\end{array}\right]$ and $B=\left[\begin{array}{rrrr}1& 3& 2& 2\\ 2& 4& 5& 1\end{array}\right]$.
From above, we know that $AB$ is defined since the number of columns in ${A}_{3×2}$ $\left(2\right)$ matches the number of rows in ${B}_{2×4}$ $\left(2\right)$.
To find $AB$, we must be sure to find the dot product between each row in $A$ and each column in $B$. So, the resulting matrix will have the same number of rows as matrix ${A}_{3×2}$ $\left(3\right)$ and the same number of columns as matrix ${B}_{2×4}$ $\left(4\right)$. It will be a $3×4$ matrix.
$\begin{array}{cccccccc}& & & & \stackrel{\to }{{b}_{1}}& \stackrel{\to }{{b}_{2}}& \stackrel{\to }{{b}_{3}}& \stackrel{\to }{{b}_{4}}\\ & & & & ↓& ↓& ↓& ↓\\ \\ \begin{array}{c}\stackrel{\to }{{a}_{1}}\to \\ \stackrel{\to }{{a}_{2}}\to \\ \stackrel{\to }{{a}_{3}}\to \end{array}& \left[\begin{array}{c}1\\ 2\\ 2\end{array}& \begin{array}{c}3\\ 4\\ 5\end{array}\right]& \cdot & \left[\begin{array}{c}1\\ 2\end{array}& \begin{array}{c}3\\ 4\end{array}& \begin{array}{c}2\\ 5\end{array}& \begin{array}{c}2\\ 1\end{array}\right]& =& \left[\begin{array}{c}\stackrel{\to }{{a}_{1}}\cdot \stackrel{\to }{{b}_{1}}\\ \stackrel{\to }{{a}_{2}}\cdot \stackrel{\to }{{b}_{1}}\\ \stackrel{\to }{{a}_{3}}\cdot \stackrel{\to }{{b}_{1}}\end{array}& \begin{array}{c}\stackrel{\to }{{a}_{1}}\cdot \stackrel{\to }{{b}_{2}}\\ \stackrel{\to }{{a}_{2}}\cdot \stackrel{\to }{{b}_{2}}\\ \stackrel{\to }{{a}_{3}}\cdot \stackrel{\to }{{b}_{2}}\end{array}& \begin{array}{c}\stackrel{\to }{{a}_{1}}\cdot \stackrel{\to }{{b}_{3}}\\ \stackrel{\to }{{a}_{2}}\cdot \stackrel{\to }{{b}_{3}}\\ \stackrel{\to }{{a}_{3}}\cdot \stackrel{\to }{{b}_{3}}\end{array}& \begin{array}{c}\stackrel{\to }{{a}_{1}}\cdot \stackrel{\to }{{b}_{4}}\\ \stackrel{\to }{{a}_{2}}\cdot \stackrel{\to }{{b}_{4}}\\ \stackrel{\to }{{a}_{3}}\cdot \stackrel{\to }{{b}_{4}}\end{array}\right]\\ \\ & A& & & & B& & & & & AB\end{array}$

4) $A=\left[\begin{array}{rr}2& 4\\ 6& 4\\ 7& 3\end{array}\right]$ and $B=\left[\begin{array}{rr}2& 1\\ 8& 5\end{array}\right]$.
What are the dimensions of $AB$?
$×$

5) $C=\left[\begin{array}{rrr}4& 3& 1\\ 6& 7& 2\end{array}\right]$ and $D=\left[\begin{array}{r}3\\ 1\\ 4\end{array}\right]$.
What are the dimensions of $CD$?
$×$

6) $A$ is a $2×3$ matrix and $B$ is a $3×4$ matrix.
What are the dimensions of $AB$?
$×$

7) $X$ is a $2×1$ matrix and $Y$ is a $1×2$ matrix.
What are the dimensions of matrix $XY$?
$×$

## Want to join the conversation?

• What is the product of AB in the top question on this page?
• Hello!

The first thing to do will be to determine the dimensions of our product matrix (I'll call it C). Because matrix A has 3 rows, and matrix B has 2 columns, matrix C will be a 3x2 matrix. 3 rows, 2 columns.

Now, the rules for matrix multiplication say that entry i,j of matrix C is the dot product of row i in matrix A and column j in matrix B. We can use this information to find every entry of matrix C. Here are the steps for each entry:

Entry 1,1: (2,4) * (2,8) = 2*2 + 4*8 = 4 + 32 = 36
Entry 2,1: (6,4) * (2,8) = 6*2 + 4*8 = 12 + 32 = 44
Entry 3,1: (7,3) * (2,8) = 7*2 + 3*8 = 14 + 24 = 38
Entry 1,2: (2,4) * (1,5) = 2*1 + 4*5 = 2 + 20 = 22
Entry 2,2: (6,4) * (1,5) = 6*1 + 4*5 = 6 + 20 = 26
Entry 3,2: (7,3) * (1,5) = 7*1 + 3*5 = 7 + 15 = 22

  36     22  44     26  38     22

I've left a link to a full tutorial on matrix multiplication at the bottom. Hope this helps!

• can i multiply more than 3 matrices?
• You could multiply as many matrices as you like, so long as the order of multiplication and the dimensions of the matrices are such that multiplication is always well-defined. The easiest way to make sure it's well-defined is to multiply a bunch of square matrices of equal dimensions.
• how do you find out the dimensions for matrices that are not defined or is it not possible
• Not possible.
A matrix that is not defined cannot exist; it has no dimensions because there is no solution to the question.
• Why (like in questions 3 & 4) is AB defined but BA is not defined? Do matrices not conform to commutative properties of multiplication?
• Correct... the commutative property of multiplication does not work with matrices. Go back and pay close attention to the section above titled: When is Matrix Multiplication Defined?". The last line states: "So, it follows that in order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix". The 2 questions you are asking about do not meet this condition which is why the multiplication is not defined.

Hope this helps.
• how can I learn the non-zero value in matrix
• A non-zero value in a matrix is where the matrix does not have a zero in it.
• How can I find the inverse of a 3*3?
• What is the point of ascertaining whether the resultant matrix from a specific ordered multiplication is DEFINED?
• Unlike multiplication of numbers or variable, the usual properties of multiplication do not always apply to multiplication of matrices. You need to learn what you can do and what you can't do.