Matrix Notation

Entry, Rows and Columns

A matrix is a way to organize data in columns and rows. A matrix is written inside brackets [ ]. Look at the picture below to see an example.
Each item in a matrix is called an entry.

Example of a Matrix

The matrix pictured below has two rows and three columns.

  • Its dimensions are 2 ×3
    • 2 rows and three columns
  • The entries of the matrix below are 2, -5, 10, -4, 19, 4.
matrix picture
picture of matrix rows
picture of matrix columns
picture of rows and columns of matrix

Dimension of Matrix

The dimensions of a matrix refer to the number of rows and columns of a given matrix. By convention the dimension of a a matrix are given by number of rows • number of columns

One way that some people remember that the notation for matrix dimensions is rows by columns (Rather than columns by rows) is by recalling a once popular-soda:

                    RC Cola	-- rows before columns!

Below, you can see two pictures of the same matrix with the rows and columns highlighted.

The dimensions of this matrix
  • dimensions: 2 × 3
  • 2 rows × 3 columns

Matrix Notation

In order to identify an entry in a matrix, we simply write a subscript of the respective entry's row followed by the column.

example of matrix notation

In matrix A on the left, we write a23 to denote the entry in the second row and the third column.

One way to remember that this notation puts rows first and columns second is to think of it like reading a book. You always read sideways first, just as you always write the rows first. To continue the analogy, when you are done reading a row in a book, your eyes move downward, just as the columns after the rows.    A23 indicates the row number first, 2, then the column number 3.

Practice Identifying Entries

Practice 1
dimensions of matrix
Step 1

What are the dimensions of the matrix below?

The dimensions of the matrix are 3 × 3 (3 rows × 3 columns)

Step 2

Identify entry G23 in the matrix G on the left.

G23 is the entry in the second row and third column: 55.

Step 3

What are the dimensions of J?

The dimensions are 3 × 5 (3 rows × 5 columns)

Practice 2
entry of a matrix
Step 1

Identify entry j34

Entry j34 is the entry in the third row and fourth column.

Step 2

Identify entry J 12

J12 is the number in the first row and the second column: -5

Practice 3
dimensions of matrix
Step 1

What are the dimensions of V below?

The dimensions of the matrix are 4 × 5 (4 rows × 5 columns)

Step 2

Identify entry v14

The entry is 31, (row 1 & column 4)

Step 3

What is the matrix notation to denote the entry in the bottom right corner, 15?

v45 denotes the entry in the fourth row and fifth column, the number 15 in the bottom right corner.

Adding and Subtracting Matrices

You can add or subtract matrices if each matrix has the same dimensions (in other words, each one needs to have exactly the same number of columns and rows).

To add or subtract matrices , you just add or subtract the corresponding entries (the entries or numbers that are in the same spot).

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Why are the same dimensions required for addition and subtraction?

Think about it: Since adding/subtracting matrices involves adding/subtracting corresponding entries. What would you do with the entries in the one matrix that do not have a corresponding entry in the other?)

Consider the example pictured up above.. Matrix #1 has one more column than Matrix #2. How would you match, let alone add, the entries of #1's column 3 with corresponding ones in #2. Well, the answer is - you can't since you cannot add matrixes unless they have the same number of rows and columns .

next to Multiplying Matrices