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Operations on matrices
We want to define operations on matrices: some (useful) ways of taking two matrices and making a new matrix.
Before we begin, a remark about $1\times 1$ matrices. These are of the form $[a_{11}]$ where $a_{11}$ is just a number. The square brackets $[\,]$ don't really matter here; they just keep the inside of a matrix in one place. So really: a $1\times 1$ matrix is just a number. This means that special cases of the operations we define will be operations on ordinary numbers. You should check that in the special case when all the matrices involved are $1\times 1$ matrices, the operations become the ordinary operations on numbers, so we are generalising the familiar operations (addition, subtraction, multiplication and so on) from numbers to matrices.
Matrix addition and subtraction
Definition of matrix addition
If $A$ and $B$ are matrices of the same size, then $A+B$ is defined to be the matrix with the same size as $A$ and $B$ so that the $(i,j)$ entry of $A+B$ is $a_{ij}+b_{ij}$, for every $i,j$.
If $A$ and $B$ are matrices of different sizes, then $A+B$ is undefined.
Example
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}+\begin{bmatrix}-2&2&0\\1&1&1\end{bmatrix}=\begin{bmatrix}-1&4&-2\\4&1&6\end{bmatrix}.\]
Example
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}+\begin{bmatrix}-2&2\\1&1\end{bmatrix}\text{ is undefined.}\]
Remarks
- For any matrices $A$ and $B$ with the same size: $A+B=B+A$. We say that matrix addition is commutative.
- For any matrices $A$, $B$ and $C$ with the same size: $(A+B)+C=A+(B+C)$. We say that matrix addition is associative.
Definition of the zero matrix
The $n\times m$ zero matrix is the $n\times m$ matrix so that every entry is $0$. We write this as $0_{n\times m}$. So \[ 0_{n\times m}=\begin{bmatrix} 0&0&\dots&0\\ 0&0&\dots&0\\ \vdots&\vdots&&\vdots\\ 0&0&\dots&0\end{bmatrix}\] where this matrix has $n$ rows and $m$ columns.
Exercise
Show that if $A$ is any $n\times m$ matrix, then \[ 0_{n\times m}+A=A=A+0_{n\times m}.\] Remember that when checking that matrices are equal, you have to check that they have the same size, and that all the entries are the same.
Definition of matrix subtraction
If $A$ and $B$ are matrices of the same size, then $A-B$ is defined to be the matrix with the same size as $A$ and $B$ so that the $(i,j)$ entry of $A-B$ is $a_{ij}-b_{ij}$, for every $i,j$.
If $A$ and $B$ are matrices of different sizes, then $A-B$ is undefined.
Example
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}-\begin{bmatrix}-2&2&0\\1&1&1\end{bmatrix}=\begin{bmatrix}3&0&-2\\2&-1&4\end{bmatrix}.\]
Example
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}-\begin{bmatrix}-2&2\\1&1\end{bmatrix}\text{ is undefined.}\]