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I have the following two matrices:

$$A=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ and }B=\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$

So I have two matrixes with different sizes. Multiple sources tell me that I can't do multiplication or addition with matrix of different sizes. So I'm a bit confused. Can I do it with these? How?

Carl Schildkraut
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kirkegaard
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  • You can multiply them in one order but not the other. Why would you expect to be able to add them? – Tobias Kildetoft Apr 13 '15 at 12:53
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    You cannot add them, as some entries of the matrix $B$ have no corresponding elements from $A$ to be added to (specifically, the elements on the third column of $B$). You can, however, multiply them. Since the size of $A$ is $2\times 2$, and the size of $B$ is $2\times 3$, the following matrix multiplications can be performed: $AB$ and $B^tA$, where $t$ denotes the matrix transpose operator. – Meshal Apr 13 '15 at 12:56

4 Answers4

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Consider you have two matrices $A$ and $B$ of orders $a_1\times a_2$ and $b_1\times b_2$ respectively.

Matrix addition/subtraction on the two matrices will be defined iff $a_1=b_1$ and $a_2=b_2$

Matrix multiplication on them is defined iff $a_2=b_1$ for $AB$ to be defined and $b_2=a_1$ for $BA$ to be defined. $AB$ will be of order $a_1\times b_2$ and $BA$ will be of order $b_1\times a_2$

Prasun Biswas
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In order to add two matrices, they must have the same dimensions, so you cannot add your matrices.

In order to multiply to matrices $M$ and $N$, the number of columns of $M$ must be equal to the number of rows of $N$. In your case, you can multiply $A\cdot B$ because the number of columns of $A$ is $2$ and the number of rows of $B$ is $2$.

You cannot multiply $B\cdot A$ because the number of columns of $B$ is $3$ and the number of rows of $A$ is $2$.

5xum
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Matrix addition and multiplication were not defined out of the blue. An $m\times n$ matrix describes a linear transformation from a vector space of dimension $n$ to a vector space of dimension $m$.

The sum of matrices corresponds to the transformation which is the sum of the two given transformations. Addition of linear transformations require the domain and range to be identical.

The product of matrices corresponds to the transformation which is the composition of the two given transformations. Composition of linear transformations require the range of the first to be the domain of the second.

Hence the constraints on the sizes. If they are not met, the result is not defined.

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So I have two matrixes with different sizes. Multiple sources tell me that I can't do multiplication or addition with matrix of different sizes.

You can't add matrixes of different sizes as stated by @meshal. Hence, A+B or B+A can't be performed. However, you can multiply them. Here's how:

$$A=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ and }B=\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$

$$AB=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ * }\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$

$$AB=\begin{pmatrix}1\cdot \:1+\left(-2\right)\left(-1\right)&1\cdot \:3+\left(-2\right)\cdot \:0&1\cdot \:2+\left(-2\right)\cdot \:2\\ 3\cdot \:1+1\cdot \left(-1\right)&3\cdot \:3+1\cdot \:0&3\cdot \:2+1\cdot \:2\end{pmatrix}$$

$$Finally, AB=\begin{pmatrix}3&3&-2\\ 2&9&8\end{pmatrix}$$

Sakib Arifin
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