Matrix
Pronunciation: /ˈmeɪ trɪks/ Explain
 plural is matrices.
 Figure 1: 2x3 matrix 

A matrix is made up of values arranged in rows and columns.
In advanced math, the values can be variables, equations or even other matrices. This
article will deal with matrices that contain only numbers.

Article Index
Matrix
Matrix Rows and Columns
Matrix Dimension
Square Matrix
Diagonal of a Square Matrix
Augmented Matrix
Matrix Element
Corresponding Elements
Equality of Matrices
Matrix Addition
Scalar Multiplication
Matrix Multiplication
Using Matrices to Solve Linear Systems
Determinant of a Matrix
Cramer's Rule
Matrix
A matrix is used to organize data. Once data is organized into a matrix, standard matrix
operations can be performed to manipulate the data. Here is an example from retail sales.
Table 1 contains sales information for the first three months of the year for a store.
Table 2 contains sales information for the second three months of the year for the same store.
1st Quarter Sales, Store #482
 January  February  March 
Fruit  $3045  $2997  $3200 
Vegetables  $4056  $4227  $4509 
Pasta  $2650  $3204  $3098 
Dairy  $5345  $4723  $4933 
Figure 2: Table of first quarter sales. 

2nd Quarter Sales, Store #482
 April  May  June 
Fruit  $3420  $3560  $3700 
Vegetables  $4716  $4850  $4900 
Pasta  $3274  $2840  $2760 
Dairy  $4769  $4799  $4873 
Figure 3: Table of second quarter sales. 

These tables 1 and 2 can be organized into two matrices:

Figure 4: Matrix corresponding to table in figure 2.



Figure 5: Matrix corresponding to table in figure 3.


Notice that none of the labels are transferred to the matrix. However, the numbers
in the rows and columns still have the same meaning. Everything in the first row
is sales numbers for fruit. Everything in the second row is sales numbers for vegetables.
Everything in the first column is the sales for the first month of the quarter.
Understanding Check
Click on the check box of the answer you think is correct.
 What is the meaning of matrix 2, row 3?
Sales of fruit.
No, since row 3 of matrix 2 corresponds to row three of table 2, row 3 represents sales of pasta.
Sales of vegetables.
No, since row 3 of matrix 2 corresponds to row three of table 2, row 3 represents sales of pasta.
Sales of pasta.
Yes, since row 3 of matrix 2 corresponds to row three of table 2, row 3 represents sales of pasta.
Sales of dairy products.
No, since row 3 of matrix 2 corresponds to row three of table 2, row 3 represents sales of pasta.
 What is the meaning of matrix 1, column 2?
January sales.
No, since column 2 of matrix 1 corresponds to column 2 of table 2, row 3 represents February sales.
February sales.
No, since column 2 of matrix 1 corresponds to column 2 of table 2, row 3 represents February sales.
May sales.
Yes, since column 2 of matrix 1 corresponds to column 2 of table 2, row 3 represents February sales.
June sales.
No, since column 2 of matrix 1 corresponds to column 2 of table 2, row 3 represents February sales.
 What is the meaning of matrix 1, column 1, row 2?
Sales of fruit in January.
No, since column 1, row 2 of matrix 1 corresponds to column 1, row 2 of table 1, this cell represents sales of vegetables in January.
Sales of vegetables in March.
No, since column 1, row 2 of matrix 1 corresponds to column 1, row 2 of table 1, this cell represents sales of vegetables in January.
Sales of fruit in April.
No, since column 1, row 2 of matrix 1 corresponds to column 1, row 2 of table 1, this cell represents sales of vegetables in January.
Sales of vegetables in May.
Yes, since column 1, row 2 of matrix 1 corresponds to column 1, row 2 of table 1, this cell represents sales of vegetables in January.
A Matrix is Organized in Columns and Rows

Matrix rows and columns.


A matrix is organized in columns and rows. A column is a single vertical arrangement of
values. A row is a single horizontal arrangement of values. A matrix can have one or more
columns and one or more rows. If the values in the matrix represent real data,
each column and each row has meaning.

Matrix Dimension
Each matrix has a dimension. Since the matrix
has 1 row and 3 columns, the dimensions of this matrix are 1x3.
Matrix  Number of Rows  Number of Columns  Dimensions 
 2  1  2x1 
 2  2  2x2 
 3  4  3x4 
Figure 6: Dimensions of Matrices 
Square Matrix
 Figure 7: A square 2x2 matrix 

A square matrix is a matrix that has the same number of rows as
columns. One property of a square matrix is that a square matrix can be multiplied by
itself. This is not true for matrices that are not square.

Diagonal of a Square Matrix
 Figure 8: The diagonal of a 3x3 matrix 

The diagonal of a square matrix A is the
elements A_{1,1}, A_{2,2},
A_{3,3}, ….
In figure 8, the diagonal is highlighted in red. Some mathematicians call the diagonal of a matrix a
main diagonal.

Augmented Matrix
 Figure 8a: An augmented matrix. 

An augmented matrix is a square matrix with a column added. An
augmented matrix is used to represent a linear system. Each column of the square matrix
represents a variable. The augmented part of the matrix represents the constant term
of each linear equation. An augmented matrix is sometimes represented using a vertical line
to separate the square matrix from the augmented part. See figure 8b.

 Figure 8b: Augmented matrix with a vertical separator. 

Matrix Element
 Figure 9: Matrix A 

Each entry in a matrix is called a matrix element. Each
element is identified by row and column in the array. The highlighted element in matrix A is
identified as A[2,1]. This means the element in the second row, first column of matrix A.

Understanding Check
Click on the check box of the answer you think is correct.
 What is the value of the element A[3,2]?
1
No. The element in row 3 (count down 3 rows) and column 2 (count over 2 columns) has a value of 4.
4
Yes. The element in row 3 (count down 3 rows) and column 2 (count over 2 columns) has a value of 4.
5
No. The element in row 3 (count down 3 rows) and column 2 (count over 2 columns) has a value of 4.
8
No. The element in row 3 (count down 3 rows) and column 2 (count over 2 columns) has a value of 4.
 What is the value of the element A[1,3]?
1
No. The element in row 1 (count down 1 row) and column 3 (count over 3 columns) has a value of 7.
3
No. The element in row 1 (count down 1 row) and column 3 (count over 3 columns) has a value of 7.
5
No. The element in row 1 (count down 1 row) and column 3 (count over 3 columns) has a value of 7.
7
Yes. The element in row 1 (count down 1 row) and column 3 (count over 3 columns) has a value of 7.
 What element of Matrix A has a value of 1?
A[1,1]
No. The element A[2,2] = 1.
A[1,2]
No. The element A[2,2] = 1.
A[2,1]
No. The element A[2,2] = 1.
A[2,2]
Yes. The element A[2,2] = 1.
Corresponding Elements
Corresponding elements in matrices are elements at the same row and column. In figure 10,
corresponding
elements of the two matrices are the same color.

Figure 10: Corresponding elements 
Equality of Matrices
Two matrices are equal if and only if two conditions are met:
 The matrices have the same number of rows and columns.
 The corresponding elements of the matrices are equal.
Examples of Matrix Equality
Example  Discussion 
 These two matrices are equal. They have the same number of rows and the same number of columns and their corresponding elements are equal. 
 These matrices are not equal. The have the same number of rows and the same number of columns, but the elements in position 1,1 are not equal. 
 These matrices are not equal. While the corresponding elements are equal, the first matrix has two columns and the second matrix has three columns. 
Table 11  Examples of matrix equality. 
Matrix Addition
Matrix addition is defined as adding
corresponding elements.
Let matrix
and matrix
To add the matrix B to C, add the value of
B[1,1] to
C[1,1],
B[1,2] to
C[1,2], and so on as follows:

Figure 12: Matrix addition 
Since each element in the first matrix must have a corresponding element in the
second matrix, only matrices with the same dimensions can be added. Addition of
matrices with different dimensions is
undefined.
This means that you can not add matrices with different dimensions.
Matrix subtraction is similar to matrix addition.
Subtract corresponding elements.

Figure 13: Matrix subtraction 
Scalar Multiplication
 Figure 14: Scalar multiplication 

In scalar multiplication, a
scalar
is a number that is multiplied by each element of the matrix. Figure 13 shows the scalar 5 being multiplied by a matrix. Each element
of the matrix is multiplied by five.

Matrix Multiplication
Matrix multiplication is defined as multiplying the elements
of each row of the first matrix by each column of the second matrix.
Figure 15 shows an example of matrix multiplication.

Figure 15: Matrix multiplication. 
To multiply two matrices, they must be compatible. This means
that the number of columns in the first matrix must be the same as the number of rows in
the second matrix. Figure 16 gives some examples of compatible and incompatible matrices.
Matrix 1  Matrix 2  Compatibility 
  Compatible. These matrices are compatible. They can be multiplied. The first matrix has 2 columns and the second matrix has 2 rows. 
  Incompatible. These matrices are incompatible. They can not be multiplied. The first matrix has 2 columns and the second matrix has 3 rows. 
  Compatible. These matrices are compatible. They can be multiplied. The first matrix has 3 columns and the second matrix has 3 rows. 
  Incompatible. These matrices are incompatible. They can not be multiplied. The first matrix has 2 columns and the second matrix has 3 rows. Note: These are the same matrices in the previous example, but have been reversed. This means that matrix multiplication is not commutative. 
Figure 16: Matrix compatibility 
It is easier to remember how to multiply matrices if you move the second matrix up. Take,
for example, the matrix multiplication problem:
The second matrix is moved up like this:
Now the columns and rows of the matrices that will be multiplied are aligned. Start with row 1
and column 1 of the matrix. The elements in the first row of the first matrix are multiplied
by the elements in the first column of the second matrix:
2·0+
(3)·1+
0·3 = 03+0 =
3.
Now multiply row 1 of matrix 1 by column 2 of matrix 2.
2·(1)+
(3)·2+
0·(3) = 2+6+0 =
4.
Now multiply row 2 of matrix 1 by column 1 of matrix 2.
2·(0)+
1·1+
(6)·3 = 0+118 =
17.
Continue in the same pattern until all the rows of the first matrix have been multiplied by
all the rows of the second matrix.
This is how the problem look laid out horizontally:
More Information
 McAdams, David E. Determinant of a Matrix. allmathwords.org. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/d/determinant.html.
Educator Resources
 Wired For Space (arrays). NASA LaRC Office of Education. 2/18/2010. http://www.archive.org/details/NasaConnectWiredForSpace/index.htm.
Cite this article as:
McAdams, David E. Matrix. 4/30/2011. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/matrix.html.
Image Credits
Revision History
3/30/2011: Added information on columns and rows. (
McAdams, David E.)
9/28/2009: Minor wording changes to improve readability. (
McAdams, David E.)
12/3/2008: Added 'More Information', link to Determinant. (
McAdams, David E.)
11/17/2008: Added matrix multiplication. (
McAdams, David E.)
3/10/2008: Initial version. (
McAdams, David E.)