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
are shown below.
A matrix is often referred to by its size or dimensions:
columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix
we look for the entry in row
. In matrix
shown below, the entry in row 2, column 3 is
A square matrix
is a matrix with dimensions
meaning that it has the same number of rows as columns. The
matrix above is an example of a square matrix.
A row matrix
is a matrix consisting of one row with dimensions
A column matrix
is a matrix consisting of one column with dimensions
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
is a rectangular array of numbers that is usually named by a capital letter:
and so on. Each entry in a matrix is referred to as
, such that
represents the row and
represents the column. Matrices are often referred to by their dimensions:
Example 1: Finding the Dimensions of the Given Matrix and Locating Entries
- What are the dimensions of matrix
- What are the entries at and
- The dimensions are
because there are three rows and three columns.
- Entry is the number at row 3, column 1, which is 3. The entry is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.
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
matrix and another
matrix, but we cannot add or subtract a
matrix and a
matrix because some entries in one matrix will not have a corresponding entry in the other matrix.
A General Note: Adding and Subtracting Matrices
of like dimensions, addition and subtraction of
will produce matrix
of the same dimension.
Matrix addition is commutative.
It is also associative.
Example 2: Finding the Sum of Matrices
Find the sum of
Add corresponding entries.
Example 3: Adding Matrix A and Matrix <>B
Find the sum of
Add corresponding entries. Add the entry in row 1, column 1,
to the entry in row 1, column 1,
. Continue the pattern until all entries have been added.
Example 4: Finding the Difference of Two Matrices
Find the difference of
We subtract the corresponding entries of each matrix.
Example 5: Finding the Sum and Difference of Two 3 x 3 Matrices
- Find the sum.
- Find the difference.
- Add the corresponding entries.
- Subtract the corresponding entries.
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