Learn how to multiply a matrix by another matrix.

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.
For example, matrix AA has 22 rows and 33 columns. The element a2,1a_{\blueD2,\goldD1} is the entry in the 2nd row\blueD{2\text{nd row}} and the 1st column\goldD{1\text{st column}} of matrix AA, or 55.
If this is new to you, we recommend that you check out our intro to matrices. You should also make sure you know how to multiply a matrix by a scalar.

What you will learn in this lesson

How to find the product of two matrices. For example, find
[1724][3352]\left[\begin{array}{rr}{1} &7 \\ 2& 4 \end{array}\right]\cdot\left[\begin{array}{rr}{3} &3 \\ 5& 2 \end{array}\right]

Scalar multiplication and matrix multiplication

When we work with matrices, we refer to real numbers as scalars.
The term scalar multiplication refers to the product of a real number and a matrix. In scalar multiplication, each entry in the matrix is multiplied by the given scalar.
In contrast, matrix multiplication refers to the product of two matrices. This is an entirely different operation. It's more complicated, but also more interesting! Let's see how it's done.
Understanding how to find the dot product of two ordered lists of numbers can help us tremendously in this quest, so let's learn about that first!

nn-tuples and the dot product

We are familiar with ordered pairs, for example (2,5)(2,5), and perhaps even ordered triples, for example (3,1,8)(3,1,8).
An nn-tuple is a generalization of this. It is an ordered list of nn numbers.
We can find the dot product of two nn-tuples of equal length by summing the products of corresponding entries.
For example, to find the dot product of two ordered pairs, we multiply the first coordinates and the second coordinates and add the results.
(2,5)(3,1)=23+51=6+5=11\begin{aligned}(\purpleC2,\greenD5)\cdot (\purpleC3,\greenC1)&=\purpleC2\cdot \purpleC3+\greenD5\cdot \greenD1\\ \\ &=6+5\\ \\&=11 \end{aligned}
Ordered nn-tuples are often indicated by a variable with an arrow on top. For example, we can let a=(3,1,8)\vec{a}=(3,1,8) and b=(4,2,3)\vec{b}=(4,2,3). The expression ab\vec{a}\cdot \vec{b} indicates the dot product of these two ordered triples and can be found as follows:
ab=(3,1,8)(4,2,3)=34+12+83=12+2+24=38\begin{aligned}\vec{a}\cdot \vec{b}&=(\purpleC 3,\greenD1,\maroonC8)\cdot (\purpleC4, \greenD2, \maroonC3 )\\\\&=\purpleC3\cdot \purpleC4+\greenD1\cdot \greenD2+\maroonC8\cdot \maroonC3\\ \\ &=12+2+24\\ \\&=38 \end{aligned}
Notice that the dot product of two nn-tuples of equal length is always a single real number.

Check your understanding

1) Let c=(4,3)\vec{c}=(4,3) and d=(3,5)\vec{d}=(3,5).
cd=\vec{c}\cdot \vec{d}=
cd=(4,3)(3,5)=43+35=12+15=27\begin{aligned}\vec{c}\cdot \vec{d}&=(\purpleC4, \greenD3) \cdot (\purpleC 3, \greenD 5)\\ \\ &=\purpleC4\cdot \purpleC 3+\greenD3\cdot \greenD5\\\\ &=12+15\\\\ &=27\end{aligned}
2) Let m=(2,5,2)\vec{m}=(2,5, -2) and n=(1,8,3)\vec{n}=(1,8,-3).
mn=\vec{m}\cdot \vec{n} =
mn=(2,5,2)(1,8,3)=21+58+(2)(3)=2+40+6=48\begin{aligned}\vec{m}\cdot \vec{n}&=(\purpleC 2, \greenD 5, \maroonC{-2})\cdot (\purpleC 1, \greenD 8, \maroonC{-3})\\\\ &=2\cdot 1+5\cdot 8+(-2)(-3)\\\\ &=2+40+6\\\\ &=48\end{aligned}

Matrices and nn-tuples

When multiplying matrices, it's useful to think of each matrix row and column as an nn-tuple.
In this matrix, row 11 is denoted r1=(6,2)\blueD{\vec{r_1}}=(6,2) and row 22 is denoted r2=(4,3)\blueD{\vec{r_2}}=(4,3).
Similarly, column 11 is denoted c1=(6,4)\goldD{\vec{c_1}}=(6,4) and column 22 is denoted c2=(2,3)\goldD{\vec{c_2}}=(2,3).

Check your understanding

3) Which of the following ordered triples is c2\vec{c_2}?
Choose 1 answer:
Choose 1 answer:
c2\vec{c_2} refers to the ordered triple in the second column of the matrix.
c2=(3,3,1)\vec{c_2}=(3,3,1)

Matrix multiplication

We are now ready to look at an example of matrix multiplication.
Given A=[1724]A=\left[\begin{array}{rr}{1} &7 \\ 2& 4 \end{array}\right] and B=[3352]B=\left[\begin{array}{rr}{3} &3 \\ 5& 2 \end{array}\right], let's find matrix C=ABC=AB.
To help our understanding, let's label the rows in matrix AA and the columns in matrix BB. We can define the product matrix, matrix CC, as shown below.
Notice that each entry in matrix CC is the dot product of a row in matrix AA and a column in matrix BB. Specifically, the entry ci,jc_{\blueD i,\goldD j} is the dot product of ai\blueD{\vec{a_i}} and bj\goldD{\vec{b_j}}.
In the image below, we see that c1,2c_{\blueD1,\goldD2} is the dot product of a1\blueD{\vec{a_1}} and b2\goldD{\vec{b_2}}.
c1,2=(1,7)(3,2)=13+72=3+14=17\begin{aligned}c_{\blueD1,\goldD2}&=(\blueD1,\blueD7)\cdot (\goldD3,\goldD2)\\ &=\blueD1\cdot \goldD3+\blueD7\cdot \goldD2\\&=3+14\\&=17 \end{aligned}
We can complete the dot products to find the complete product matrix:
C=[38172614]C=\left[\begin{array}{rr}{38} &17 \\ 26& 14 \end{array}\right]
c1,1=a1b1=(1,7)(3,5)=13+75=3+35=38\begin{aligned}c_{\blueD1,\goldD1}&=\blueD{\vec{a_1}}\cdot \goldD{\vec{b_1}}\\ &=(\blueD1,\blueD7)\cdot (\goldD3,\goldD5)\\ &=\blueD1\cdot \goldD3+\blueD7\cdot \goldD5\\&=3+35\\&=38 \end{aligned}
c2,1=a2b1=(2,4)(3,5)=23+45=6+20=26\begin{aligned}c_{\blueD2,\goldD1}&=\blueD{\vec{a_2}}\cdot \goldD{\vec{b_1}}\\ &=(\blueD2,\blueD4)\cdot (\goldD3,\goldD5)\\ &=\blueD2\cdot \goldD3+\blueD4\cdot \goldD5\\&=6+20\\&=26 \end{aligned}
c2,2=a2b2=(2,4)(3,2)=23+42=6+8=14\begin{aligned}c_{\blueD2,\goldD2}&=\blueD{\vec{a_2}}\cdot \goldD{\vec{b_2}}\\ &=(\blueD2,\blueD4)\cdot (\goldD3,\goldD2)\\ &=\blueD2\cdot \goldD3+\blueD4\cdot \goldD2\\&=6+8\\&=14 \end{aligned}

Check your understanding

4) C=[2152]C=\left[\begin{array}{rr}{2} &1 \\ 5& 2 \end{array}\right] and D=[1436]D=\left[\begin{array}{rr}{1} &4 \\ 3& 6 \end{array}\right].
Let F=CDF=C\cdot D.
a) Which of the following is f2,1f_{2,1}?
Choose 1 answer:
Choose 1 answer:
The entry f2,1f_{\blueD 2,\goldD 1} is the dot product of 2nd row in C\blueD{2\text{nd row in }C} and the 1st column in D\goldD{1\text{st column in }D}.
Since C=[2152]C=\left[\begin{array}{rr}{2} &1 \\ \blueD5& \blueD2 \end{array}\right] and D=[1436]D=\left[\begin{array}{rr}{\goldD1} &4 \\ \goldD3& 6 \end{array}\right], we have:
f2,1=(5,2)(1,3)=51+23=11\begin{aligned}f_{\blueD2,\goldD1}&=\blueD{(5,2)}\cdot \goldD{(1,3)}\\ \\ &=5\cdot 1+2\cdot 3\\ \\ &=11 \end{aligned}
b) Find FF.
F=F=
Let's start by labeling the rows in matrix CC and the columns in matrix DD.
We can find the product, matrix FF, as follows:
F=[2152][1436]=[c1d1c1d2c2d1c2d2]=[21+1324+1651+2354+26]=[5141132]\begin{aligned}F&=\left[\begin{array}{rr}{2} & {1} \\ 5 & 2\end{array}\right]\left[\begin{array}{rr} 1 & 4 \\ {3} & 6\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{\vec{c_1}\cdot \vec{d_1}} & {\vec{c_1}\cdot \vec{d_2}} \\ \vec{c_2}\cdot \vec{d_1} & \vec{c_2}\cdot \vec{d_2}\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{2\cdot 1+1\cdot 3} & {2\cdot 4+1\cdot 6} \\ 5\cdot 1+2\cdot 3 &5\cdot4+2\cdot 6\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{5} & {14} \\ 11 & 32\end{array}\right]\end{aligned}
5) X=[4123]X=\left[\begin{array}{rr}{4} &1 \\ 2& 3 \end{array}\right] and Y=[2854]Y=\left[\begin{array}{rrr}{2} &8 \\ 5& 4 \end{array}\right].
Find Z=XYZ=X\cdot Y.
Z=Z=
Let's start by labeling the rows in matrix XX and the columns in matrix YY.
We can find the product, matrix ZZ, as follows:
Z=[4123][2854]=[x1y1x1y2x2y1x2y2]=[42+1548+1422+3528+34]=[13361928]\begin{aligned}Z&=\left[\begin{array}{rr}{4} &1 \\ 2& 3 \end{array}\right]\left[\begin{array}{rrr}{2} &8 \\ 5& 4 \end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{\vec{x_1}\cdot \vec{y_1}} & {\vec{x_1}\cdot \vec{y_2}} \\ \vec{x_2}\cdot \vec{y_1} & \vec{x_2}\cdot \vec{y_2}\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{4\cdot 2+1\cdot 5} & {4\cdot 8+1\cdot 4} \\ 2\cdot 2+3\cdot 5 &2\cdot8+3\cdot 4\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{13} & {36} \\ 19 &28\end{array}\right]\end{aligned}
6) M=[283541]M=\left[\begin{array}{rrr}{2} &8 &3 \\ 5& 4&1 \end{array}\right] and N=[416324]N=\left[\begin{array}{rr}{4} &1 \\ 6& 3\\2&4 \end{array}\right].
Let P=MNP=M\cdot N.
a) Which of the following is p1,2p_{1,2}?
Choose 1 answer:
Choose 1 answer:
The entry p1,2p_{\blueD 1,\goldD 2} is the dot product of 1st row in M\blueD{1\text{st row in }M} and the 2nd column in N\goldD{2\text{nd column in }N}.
Since M=[283541]M=\left[\begin{array}{rrr}{\blueD2} &\blueD8 &\blueD3 \\ 5& 4&1 \end{array}\right] and N=[416324]N=\left[\begin{array}{rr}{4} &\goldD1 \\ 6& \goldD3\\2&\goldD4 \end{array}\right] , we have:
p1,2=(2,8,3)(1,3,4)=21+83+34=2+24+12=38\begin{aligned}p_{\blueD1,\goldD2}&=\blueD{(2,8,3)}\cdot \goldD{(1,3,4)}\\ \\ &=2\cdot 1+8\cdot 3+3\cdot 4\\ \\ &=2+24+12\\ \\ &=38 \end{aligned}
b) Find PP.
P=P=
Let's start by labeling the rows in matrix MM and the columns in matrix NN.
We can find the product, matrix PP, as follows:
P=[283541][416324]=[m1n1m1n2m2n1m2n2]=[24+86+3221+83+3454+46+1251+43+14]=[62384621]\begin{aligned}P&=\left[\begin{array}{rrr}{2} &8 &3 \\ 5& 4&1 \end{array}\right]\left[\begin{array}{rr}{4} &1 \\ 6& 3\\2&4 \end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{\vec{m_1}\cdot \vec{n_1}} & {\vec{m_1}\cdot \vec{n_2}} \\ \vec{m_2}\cdot \vec{n_1} & \vec{m_2}\cdot \vec{n_2}\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{2\cdot 4+8\cdot 6+3\cdot 2} & {2\cdot 1+8\cdot 3+3\cdot4} \\ 5\cdot 4+4\cdot 6+1\cdot 2 &5\cdot1+4\cdot 3+1\cdot 4\end{array}\right]\\\\\\ &=\left[\begin{array}{rr}{62} & {38} \\ 46&21\end{array}\right]\end{aligned}

Why is matrix multiplication defined this way?

Up until now, you may have found operations with matrices fairly intuitive. For example when you add two matrices, you add the corresponding entries.
But things do not work as you'd expect them to work with multiplication. To multiply two matrices, we cannot simply multiply the corresponding entries.
If this troubles you, we recommend that you take a look at the following articles, where you will see matrix multiplication being put to use.
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