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Chapter 2: The algebra of matrices
Definition
An $n\times m$ matrix is a grid of numbers with $n$ rows and $m$ columns: \[ A=\begin{bmatrix}a_{11}&a_{12}&\dots&a_{1m}\\a_{21}&a_{22}&\dots&a_{2m}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\dots&a_{nm}\end{bmatrix}\]
The $(i,j)$ entry of a matrix $A$ is $a_{ij}$, the number in row $i$ and column $j$ of $A$.
Examples
- If $B=\begin{bmatrix} 1&3&5\\7&-20&14\end{bmatrix}$, then $B$ is a $2\times 3$ matrix, and the $(1,3)$ entry of $B$ is $b_{13}=5$, then $(2,1)$ entry is $b_{21}=7$, etc.
- $\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}$ is a $5\times 1$ matrix. A matrix like this with one column is called a column matrix.
- $\begin{bmatrix}3&2&4&0&-1\end{bmatrix}$ is a $1\times 5$ matrix. A matrix like this with one row is called a row matrix.
Even though the row matrix and the column matrix above have the same entries, they have a different “shape”, or “size”, so we must think of them has being different matrices. Let's give the definitions to make this precise.
Definition
Two matrices $A$ and $B$ have the same size if they have the same number of rows, and they have the same number of columns.
Definition
Two matrices $A$ and $B$ are said to be equal if both of the following conditions hold:
- $A$ and $B$ have the same size; and
- every entry of~$A$ is equal to the corresponding entry of~$B$; in other words, for every $(i,j)$ so that $A$ and $B$ have an $(i,j)$ entry, we have $a_{ij}=b_{ij}$.