Matrix Multiplication
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Matrix Multiplication
\(A\in\mathbb R^{m\times n}\) and \(B\in\mathbb R^{n\times p}\)
AB=\left(
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{array}
\right)
\left(
\begin{array}{cccc}
b_{11} & b_{12} & \cdots & b_{1p} \\
b_{21} & b_{22} & \cdots & b_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1} & b_{n2} & \cdots & b_{np} \\
\end{array}
\right)
=
\left(
\begin{array}{cccc}
\sum_{i}a_{1i} b_{i1} & \sum_{i}a_{1i} b_{i2}& \cdots & \sum_{i}a_{1i} b_{ip}\\
\sum_{i}a_{2i} b_{i1}& \sum_{i}a_{2i}b_{i2}& \cdots & \sum_{i}a_{2i} b_{ip}\\
\vdots & \vdots & \ddots & \vdots \\
\sum_{i}a_{mi} b_{i1}& \sum_{i}a_{mi} b_{i2} & \cdots & \sum_{i}a_{mi} b_{ip}\\
\end{array}
\right)
To find one entry of the product matrix,
Matrix Multiplication
\(A\in\mathbb R^{m\times n}\) and \(B\in\mathbb R^{n\times p}\)
AB=\left(
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{array}
\right)
\left(
\begin{array}{cccc}
b_{11} & b_{12} & \cdots & b_{1p} \\
b_{21} & b_{22} & \cdots & b_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1} & b_{n2} & \cdots & b_{np} \\
\end{array}
\right)
=
\left(
\begin{array}{cccc}
\sum_{i}a_{1i} b_{i1} & \sum_{i}a_{1i} b_{i2}& \cdots & \sum_{i}a_{1i} b_{ip}\\
\sum_{i}a_{2i} b_{i1}& \sum_{i}a_{2i}b_{i2}& \cdots & \sum_{i}a_{2i} b_{ip}\\
\vdots & \vdots & \ddots & \vdots \\
\sum_{i}a_{mi} b_{i1}& \sum_{i}a_{mi} b_{i2} & \cdots & \sum_{i}a_{mi} b_{ip}\\
\end{array}
\right)
multiply the corresponding entries in a row and a column,
Matrix Multiplication
\(A\in\mathbb R^{m\times n}\) and \(B\in\mathbb R^{n\times p}\)
AB=\left(
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{array}
\right)
\left(
\begin{array}{cccc}
b_{11} & b_{12} & \cdots & b_{1p} \\
b_{21} & b_{22} & \cdots & b_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1} & b_{n2} & \cdots & b_{np} \\
\end{array}
\right)
=
\left(
\begin{array}{cccc}
\sum_{i}a_{1i} b_{i1} & \sum_{i}a_{1i} b_{i2}& \cdots & \sum_{i}a_{1i} b_{ip}\\
\sum_{i}a_{2i} b_{i1}& \sum_{i}a_{2i}b_{i2}& \cdots & \sum_{i}a_{2i} b_{ip}\\
\vdots & \vdots & \ddots & \vdots \\
\sum_{i}a_{mi} b_{i1}& \sum_{i}a_{mi} b_{i2} & \cdots & \sum_{i}a_{mi} b_{ip}\\
\end{array}
\right)
then add the products.
Matrix Multiplication
\(A\in\mathbb R^{m\times n}\) and \(B\in\mathbb R^{n\times p}\)
AB=\left(
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{array}
\right)
\left(
\begin{array}{cccc}
b_{11} & b_{12} & \cdots & b_{1p} \\
b_{21} & b_{22} & \cdots & b_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1} & b_{n2} & \cdots & b_{np} \\
\end{array}
\right)
=
\left(
\begin{array}{cccc}
\sum_{i}a_{1i} b_{i1} & \sum_{i}a_{1i} b_{i2}& \cdots & \sum_{i}a_{1i} b_{ip}\\
\sum_{i}a_{2i} b_{i1}& \sum_{i}a_{2i}b_{i2}& \cdots & \sum_{i}a_{2i} b_{ip}\\
\vdots & \vdots & \ddots & \vdots \\
\sum_{i}a_{mi} b_{i1}& \sum_{i}a_{mi} b_{i2} & \cdots & \sum_{i}a_{mi} b_{ip}\\
\end{array}
\right)
The formula may seem complicated at first,
so let's work through an example to see how it works.
A =
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
Example
B=
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
3\times 2
2\times 3
Number of columns of \(A\) = Number of rows of \(B\)
First, check the dimensions of both matrices!
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
3\times (-2)
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
The result should be a \(3\times3\) matrix!
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\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
3\times (-2)
+\,(-2)\times 4
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
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?
?
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
3\times (-2)
+\,(-2)\times 4
2\times(-2)
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
?
?
?
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
3\times (-2)
+\,(-2)\times 4
2\times(-2)
+\,4\times 4
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
?
?
?
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
3\times (-2)
+\,(-2)\times 4
2\times(-2)
+\,4\times 4
1\times(-2)
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
?
?
?
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
3\times (-2)
+\,(-2)\times 4
2\times(-2)
+\,4\times 4
1\times(-2)
+\,(-3)\times 4
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
?
?
?
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
-14
12
-14
3\times 1
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
3\times 1
+\,(-2)\times1
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
3\times 1
+\,(-2)\times1
2\times 1
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
3\times 1
+\,(-2)\times1
2\times 1
+\,4\times1
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
3\times 1
+\,(-2)\times1
2\times 1
+\,4\times1
1\times 1
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
3\times 1
+\,(-2)\times1
2\times 1
+\,4\times1
1\times 1
+\,(-3)\times 1
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
?
?
?
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
1
6
-2
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
1
6
-2
3\times 3
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
1
6
-2
3\times 3
+\,(-2)\times6
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
1
6
-2
3\times 3
+\,(-2)\times6
2\times 3
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
1
6
-2
3\times 3
+\,(-2)\times6
2\times 3
+\,4\times 6
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
12
-14
1
6
-2
3\times 3
+\,(-2)\times6
2\times 3
+\,4\times 6
1\times 3
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
12
-14
1
6
-2
3\times 3
+\,(-2)\times6
2\times 3
+\,4\times 6
1\times 3
+\,(-3)\times 6
\left(
\begin{array}{ccc}
-2 & 1 & 3 \\
4 & 1 & 6 \\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -2 \\
2 & 4 \\
1 & -3 \\
\end{array}
\right)
A
B
=
=
\left(
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right.
\left.
\begin{array}{ccc}
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\quad & \quad & \quad \\
\end{array}\right)
-14
12
-14
1
6
-2
-3
30
-15
✅
Made by
Juan Carlos Ponce Campuzano
School of Environment and Science