Demystifying Matrices Help

Introduction to Matrices

You may think you know about the matrix; however, in algebra, the matrix is not another world where you can perform magical stunts and fight bad guys.

In algebra, matrices are rectangular arrays of numbers. Look at the example of a 2-by-2 matrix:

Rows and Columns

The brackets indicate that it's a matrix. The horizontal rows are called the rows and the vertical columns are simply called columns. The numbers that appear are called the entries or elements of the matrix.

Matrices are not limited to two rows by two columns, though. Look at the following example.

This matrix is a 2-by-4 matrix; it has two rows and four columns.

rows: [m n o p] and [q r s t]

columns:

m is the 1st row, 1st column entry.

n is the 1st row, 2nd column entry.

o is the 1st row, 3rd column entry.

p is the 1st row, 4th column entry.

q is the 2nd row, 1st column entry.

r is the 2nd row, 2nd column entry.

s is the 2nd row, 3rd column entry.

t is the 2nd row, 4th column entry.

To perform operations on matrices, there are a few basic rules to follow.

Adding and Subtracting Matrices

Adding Matrices

To add two matrices of the same kind, simply add the corresponding entries.

To add matrices, both must have the same number of rows and columns. Be careful to add the same entry from both matrices.

Because matrices are added by adding corresponding entries, matrix addition is commutative and associative. In short, if A, B, and C are matrices:

A + B = B + A

And A + (B + C) = (A + B) + C

Tip: (–1)A = –A. Do you recognize that –A is an additive inverse of A? According to the additive inverse property, A + –A = 0.

Subtracting Matrices

Subtraction follows the same rule as addition.

Multiplying Matrices

If the columns in the first matrix are the same as the rows of the second matrix, you carn multiply two matrices. You'll get a matrix that is the row of the first matrix by the column of the second matrix. Note: You cannot multiply them in reverse order.

To multiply a matrix with a real number, multiply each element with this real number.

Special Matrices

Row Matrices

Some matrices consist of a single row:

A = [2 5 1 7 0]

This row matrix would be referred to as a 1-by-5 matrix.

Column Matrices

Other matrices consist of a single column. Look at the following column matrix. It would be called a 5-by-1 matrix.

B =

If the number of entries in A is the same as the elements in B, the product AB of A and B is the number obtained by pairing each entry of A with the corresponding element of B, multiplying these pairs, and adding the resulting products.

Example

Find practice problems and solutions for these concepts at Demystifying Matrices Practice Problems.