Matrix

Pronunciation: /ˈmeɪ trɪks/ ?

-- 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 Dimension
Square Matrix
Diagonal of a Square Matrix
Augmented Matrix
Matrix Element
Corresponding Elements
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.

1. 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.
2. 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.
3. 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.

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 A1,1, A2,2, A3,3, …. 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.

1. 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.
2. 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.
3. 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

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 11: 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 12: Matrix subtraction

Scalar Multiplication

Figure 13: 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 14 shows an example of matrix multiplication.

Figure 14: 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 columns in the second matrix. Figure 15 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 15: 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 = 0-3+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+1-18 = -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. Determinant of a Matrix. allmathwords.org. Life is a Story Problem.org. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Determinant%20of%20a%20Matrix.
• McAdams, David. Cramer's Rule. allmathwords.org. Life is a Story Problem.org. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Cramer's%20Rule.
• McAdams, David. Zero Matrix. allmathwords.org. Life is a Story Problem.org. 2009-09-28. http://www.allmathwords.org/article.aspx?lang=en&id=Zero%20Matrix.

Educator Resources

• Wired For Space (arrays). NASA LaRC Office of Education. 2010-02-18. http://www.archive.org/details/NasaConnect-WiredForSpace.

Cite this article as:

Matrix. 2009-09-28. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/m/matrix.html.

Translations

• Chasque para ver este artículo en español.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem.org and are licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Revision History

2009-09-28: Minor wording changes to improve readability. (McAdams, David.)
2008-12-03: Added 'More Information', link to Determinant (McAdams, David.)
2008-11-17: Added matrix multiplication (McAdams, David.)
2008-03-10: Initial version (McAdams, David.)