How to find matrix dimensions and entries

A matrix is a rectangular arrangement of numbers. Once we know how to count its rows and columns, we can describe its size and point to any entry in it. We’ll use the same row-first order for both.

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Count the rows, then the columns

$A=\left[\begin{array}{rrr}7& 3& 4\\ 1& 6& 1\\ 2& 2& 3\end{array}\right]$

Matrix $A$ has three rows and three columns, so it’s $3\times 3$. A matrix with the same number of rows and columns is square.

$B=\left[\begin{array}{rrrrr}1& 9& 0& 0& 2\end{array}\right]$

$C=\left[\begin{array}{r}0\\ 0\\ 1\end{array}\right]$

Matrix $B$ is $1\times 5$ and matrix $C$ is $3\times 1$. A row goes across and a column goes down.

Use the subscripts to find an entry

The first subscript tells us which row to use. The second tells us which column. For $a_{2,3}$, go to row 2 of $A$ and then read the third entry in that row.

Follow row 2 across and column 3 down. They meet at the entry 1.

That gives $a_{2,3}=1$. In the same way, $b_{1,4}=0$ and $c_{3,1}=1$.

Read an entry from a symbolic matrix

$K=\left[\begin{array}{rrr}a& b& c\\ d& e& f\end{array}\right]$

This matrix is $2\times 3$. Its first-row, third-column entry is $k_{1,3}=c$, while its second-row, first-column entry is $k_{2,1}=d$. The first subscript always chooses the row.

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