Learning Outcomes
- Determine the dimensions of a matrix.
- Add and subtract two matrices.
To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ) and are usually named with capital letters. For example, three matrices named A,B,A,B, and CC are shown below.
A=[1234],B=[1270−56782],C=[−103321]A=[1234],B=⎡⎢⎣1270−56782⎤⎥⎦,C=⎡⎢⎣−103321⎤⎥⎦
Describing Matrices
A matrix is often referred to by its size or dimensions: m × n m × n indicating mm rows and nn columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix AA identified as aij,aij, we look for the entry in row i,i, column jj. In matrix AA shown below, the entry in row 2, column 3 is a23a23.
A=[a11a12a13a21a22a23a31a32a33]A=⎡⎢⎣a11a12a13a21a22a23a31a32a33⎤⎥⎦
A square matrix is a matrix with dimensions n × n, n × n, meaning that it has the same number of rows as columns. The 3×33×3 matrix above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions 1 × n1 × n.
[a11a12a13][a11a12a13]
A column matrix is a matrix consisting of one column with dimensions m × 1m × 1.
[a11a21a31]⎡⎢⎣a11a21a31⎤⎥⎦
A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.
A General Note: Matrices
A matrix is a rectangular array of numbers that is usually named by a capital letter: A,B,C,A,B,C, and so on. Each entry in a matrix is referred to as aijaij, such that ii represents the row and jj represents the column. Matrices are often referred to by their dimensions: m×nm×n indicating mm rows and nn columns.
Example: Finding the Dimensions of the Given Matrix and Locating Entries
Given matrix A:A:
- What are the dimensions of matrix A?A?
- What are the entries at a31a31 and a22?a22?
A=[21024731−2]A=⎡⎢⎣21024731−2⎤⎥⎦
Try it
Adding and Subtracting Matrices
We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.
In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a 3 × 3 3 × 3 matrix and another 3 × 3 3 × 3 matrix, but we cannot add or subtract a 2 × 3 2 × 3 matrix and a 3 × 3 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.
A General Note: Adding and Subtracting Matrices
Given matrices AA and BB of like dimensions, addition and subtraction of AA and BB will produce matrix CC or matrix DD of the same dimension.
A+B=C such that aij+bij=cijA+B=C such that aij+bij=cij
A−B=D such that aij−bij=dijA−B=D such that aij−bij=dij
Matrix addition is commutative.
A+B=B+AA+B=B+A
It is also associative.
(A+B)+C=A+(B+C)(A+B)+C=A+(B+C)
Example: Finding the Sum of Matrices
Find the sum of AA and BB given
A=[abcd] and B=[efgh]A=[abcd] and B=[efgh]
Example: Adding Matrix A and Matrix B
Find the sum of AA and BB.
A=[4132] and B=[5907]A=[4132] and B=[5907]
Example: Finding the Difference of Two Matrices
Find the difference of AA and BB.
A=[−2301] and B=[8154]A=[−2301] and B=[8154]
Example: Finding the Sum and Difference of Two 3 x 3 Matrices
Given AA and B:B:
- Find the sum.
- Find the difference.
A=[2−10−21412104−22] and B=[610−20−12−4−52−2]A=⎡⎢⎣2−10−21412104−22⎤⎥⎦ and B=⎡⎢⎣610−20−12−4−52−2⎤⎥⎦
Try It
Add matrix A and matrix B.
A=[26101−3] and B=[3−215−43]
Contribute!
Candela Citations
- Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
- College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2
- Question ID 6388. Authored by: Dow,David D.. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL
- Precalculus. Authored by: OpenStax College. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution