To start off our introduction to matrices, we will first show you that a matrix is nothing but a convenient way to organize data with rows and columns.
Suppose you have a business selling T-shirts and pants. The table below shows the number of items sold for 5 days.
Monday | Tuesday | Wednesday | Thursday | Friday | |
T-shirt | 8 | 1 | 5 | 0 | 15 |
Pants | 1 | 6 | 10 | 4 | 0 |
You may re-organize the number of T-shirts and pants sold from Monday through Friday using the matrix below.
As you can see, a matrix is a rectangular array of numbers in rows and columns.
In general, we can write a matrix with m rows and n columns as shown below.
$$ A = [a_{ij}] = \begin{bmatrix} a_{11} & a_{12} & a_{13} & . . . & a_{1n}\\ a_{21} & a_{22} & a_{23} & . . . & a_{2n}\\ a_{31} & a_{32} & a_{33} & . . . & a_{3n}\\ . & . & . & & .\\ . & . & . & & .\\ . & . & . & & .\\ a_{m1} & a_{m2} & a_{m3} & . . . & a_{mn}\\ \end{bmatrix} $$
We use a capital letter such as A, B, or C to represent or name a matrix as shown above.
We use lower-case letters with double subscripts to denote the entries of the matrix. For example, a_{36} read " a sub three six, " indicates the entry in the third row and the sixth column.
A general term is represented by a_{ij} and a_{ij} indicates the entry in ith row and jth column.
Going back to the example above about T-shirts and pants, we see that the matrix above has 2 rows and 5 columns and we say the matrix is a 2 × 5 matrix. That is, the order or dimensions of the matrix is 2 × 5. Learn more about writing the dimensions of a matrix.
2 × 5 is read 2 by 5 and it does not mean multiplication!
Notice that the number of rows is listed first and that is the way it goes with matrices.
If 4 × 3 represent the dimensions of a matrix, the matrix has 4 rows and 3 columns.
Each number in a matrix is a matrix element. We like to locate the position of elements when working with matrices.
To locate elements for matrix A, use a lower case letter and a subscript with two numbers.
The number on the left of the subscript represents the row the element is located.
The number on the right of the subscript represents the column the element is located.
a_{23} is the element in the second row and third column.
a_{23} = 10
a_{15} is the element in the first row and fifth column.
a_{15} = 15
Don't let the 15 confuse you. The number in the first row and fifth column does not have to be 15. It could be anything!
Now, let's us re-organize the information above about selling T-shirts and pants!
T-shirt | Pants | |
Monday | 8 | 1 |
Tuesday | 1 | 6 |
Wednesday | 5 | 10 |
Thursday | 0 | 4 |
Friday | 15 | 0 |
The dimensions of matrix B is 5 × 2.
Important concept:
Although matrices A and B are describing the same situation, the matrices are not equal!
Two matrices are equal when they have the same dimensions and equal corresponding elements.
Simply put, write down a matrix and then write down again the exact same matrix. These two matrices are equal!
A square matrix is a matrix with the same number of columns as rows. Matrices C, D, E, and F are square matrices.
$$ C = \begin{bmatrix} a & b\\ c & d\\ \end{bmatrix} D = \begin{bmatrix} 5 & -1\\ 4 & 0\\ \end{bmatrix} E = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\\ \end{bmatrix} $$ $$ F = \begin{bmatrix} 0 & 1 & 0\\ 12 & -1 & 6\\ -2 & 0 & 2\\ \end{bmatrix} $$In a rectangular matrix, the number of rows is not equal to the number of columns.
$$ \begin{bmatrix} 2 & 0 & -2 & 8 & 0\\ 5 & 6 & 0 & -6 & 1\\ -7 & 3 & 20 & 1 & -9\\ \end{bmatrix} $$A row matrix has only a single row and at least 2 columns.
$$ \begin{bmatrix} 8 & -4 & 3 & 1 & 2\\ \end{bmatrix} $$A column matrix has only a single column and at least 2 rows.
$$ \begin{bmatrix} 4\\ 2 \\ -6\\ 0\\ 1\\ \end{bmatrix} $$A singleton matrix has only one element. A singleton matrix has only 1 row and 1 column.
$$ \begin{bmatrix} 745\\ \end{bmatrix} $$A diagonal matrix is a square matrix with any number on the main diagonal and 0's elsewhere.
$$ \begin{bmatrix} 8 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 4 & 0 & 0\\ 0 & 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{bmatrix} $$An identity matrix is a square matrix with 1's on the main diagonal and 0's elsewhere. An identity matrix is denoted by
$$ I = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{bmatrix} $$All elements of an m × n zero matrix are zero. The following matrix is a zero matrix.
$$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ \end{bmatrix} $$To add matrices A and B with the same dimensions, just add the corresponding entries or elements.
$$ A = \begin{bmatrix} a & b & c\\ d & e & f\\ \end{bmatrix} B = \begin{bmatrix} g & h & i\\ j & k & l\\ \end{bmatrix} $$Matrix subtraction
If two matrices A and B have the same dimensions, then A - B = A + (-B)
The product of a matrix and a number is a scalar product. The scalar product of a matrix A and a number k is the matrix kA. We find kA by multiplying each element of A by the number k. The number k is called scalar.
$$ A = \begin{bmatrix} 2 & -3\\ 1 & 0\\ \end{bmatrix} $$Find the scalar product 4A if A is equal to the matrix immediately above
$$ 4A = 4\begin{bmatrix} 2 & -3\\ 1 & 0\\ \end{bmatrix} = \begin{bmatrix} 4(2) & 4(-3)\\ 4(1) & 4(0)\\ \end{bmatrix} $$Starting from top to bottom, the "transpose" of a matrix is found by turning each row of the matrix into a column.
The first row becomes the first column
The second row becomes the second column
And so on ...
Suppose A is a matrix. The letter T is used as a superscript for A so that A^{T} is the transpose of A
$$ A = \begin{bmatrix} 8 & 1 & 5 & 0 & 15\\ 1 & 6 & 10 & 4 & 0\\ \end{bmatrix} A^T = \begin{bmatrix} 8 & 1\\ 1 & 6 \\ 5 & 10\\ 0 & 4\\ 15 & 0\\ \end{bmatrix} $$Let A = [a_{ij}] be an m × n matrix and let B = [b_{ij}] be an n × p matrix.
The product AB = [c_{ij}] is an m × p matrix, where c_{ij} = a_{i1} × b_{1j} + a_{i2} × b_{2j} + a_{i3} × b_{2j} + ... + a_{in} × b_{nj}
To find the entry cij in AB, just multiply the entries in row i of matrix A by the entries in column j of matrix B.
$$ A = \begin{bmatrix} 4 & 1\\ 3 & 1\\ \end{bmatrix} B = \begin{bmatrix} 1 & -1\\ -3 & 4\\ \end{bmatrix} $$Perhaps you recall that for every nonzero real number x, there is a multiplicative inverse 1/x or x^{-1}, such that x(x^{-1}) = x^{-1}(x) = 1. You can do something similar to find the inverse of a matrix.
Inverse of a matrix
For an n × n matrix A, the inverse of the matrix A^{-1} exists if A × A^{-1} = A^{-1} × A = I
I is the identity matrix.
In the previous section about matrix multiplication, matrix B is the inverse of matrix A.
Continue this introduction to matrices with some related topics: