Introduction to Matrices

(Notation & Terminology)

A matrix is a rectangular array of numbers, written between a large pair of parentheses or square brackets, and usually named by a capital letter. For example: $A = \begin{pmatrix} 2 & -1 & 0 \\ 3 & 5 & 6\end{pmatrix}, \quad B = \begin{bmatrix} 2 & 5 \\ -1 & 0 \end{bmatrix}$ $C = \begin{pmatrix} 2 & 3 & -1 \\ 0 & 4 & 5 \\ 1 & 7 & -8 \end{pmatrix}$ are all matrices.

Whether you write a matrix between square brackets or large parenthese makes no difference whatsoever, both notations are widely accepted.

Order (size) of a Matrix

Given a matrix, its order, or size, is written: $m\times n$ where:

• $$m$$ is the number of rows the matrix has
• $$n$$ is the number of columns the matrix has
If a matrix $$A$$ is of order $$m\times n$$ we'll often indicate this using notation: $A_{m,n}$ For example, consider the matrices $$A$$ and $$B$$ shown here: $A = \begin{pmatrix} 2 & 1 & -5 \\ 0 & 7 & 8 \end{pmatrix} \quad B = \begin{pmatrix} -1 & 0 & 7 \\ 6 & 2 & -3 \\ -5 & 1 & 9 \end{pmatrix}$ We say that:
• $$A$$ is a $$2\times 3$$ "two by three" matrix, which we can write $$A_{2,3}$$
• $$B$$ is a $$3\times 3$$ "three by three" matrix, which we can write $$B_{3,3}$$

Square Matrices

If a matrix has same number of rows as columns then we say it is a square matrix. We'll come across square matrices a lot in this course. Here are a few examples: $A_2 = \begin{pmatrix} 3 & 4 \\ -1 & 5 \end{pmatrix}, \quad B_3 = \begin{pmatrix} 1 & -5 & 3 \\ -4 & 2 & 6 \\ 0 & 8 & 9\end{pmatrix}$ $C_4 = \begin{pmatrix}1 & 0 & -1 & 1\\ 3 & 5 & 7 & -9\\ -2 & 4 & -6 & 8 \\ 1 & -1 & 0 & 1\end{pmatrix}$ Notice that when dealing with square

Index Notation

To refer to specific entries inside a matrix we'll often use index notation. With index notation we can refer to any specific entry using the notation: $a_{ij}$ where:

• $$i$$ refers to the row the entry is in
• $$j$$ refers to the column the entry is in
For instance, given the $$3\times 3$$ matrix $$A$$ defined as: $A = \begin{pmatrix}3 & 0 & -1 \\ 2 & 7 & 5 \\ -4 & 6 & 1 \end{pmatrix}$ we can refer to the entry $$7$$, which is on the second row and in the second column, as well as the entry $$6$$ on the thord row and in the second column by writing: $a_{22} = 7, \quad a_{32} = 6$

Generic Matrix Notation (for Square Matrices)

When working with matrices it's important to be familiar/comfortable with matrix notation. In particular when deriving/learning formula we'll work with generic matrices.

$$2\times 2$$ Matrices

When working with $$2\times 2$$ matrices we'll often use the following generic matrix $$A$$: $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $$a$$, $$b$$, $$c$$ and $$d$$ can be any real numbers.

$$3\times 3$$ Matrices

When working with $$3\times 3$$ matrices we'll need to use index notation, $$a_{ij}$$ for each of the entries in the matrix. The typical matrix we'll use is: $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$

$$4\times 4$$ Matrices and more

When working $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}$

Some "Special" Matrices

Zero Matrix

A zero matrix, also called null matrix, has all entries equal to $$0$$, and is often referred to with the notation $$0_{m,n}$$. Here are some zero matrices: $0_{2,2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, \quad 0_{1,3} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ $0_{3,3} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

Row Matrix

A row matrix only has one row: $A_{1,n} = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{pmatrix}$ Here are some examples of row matrices: $A = \begin{pmatrix} -2 & 1 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & -6 \end{pmatrix}$ Note: row matrices can be thought of as row vectors, in fact we'll often treat them that way as we learn more about matrices.

Column Matrix

A column matrix only has one column: $A_{m,1} = \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix}$ Here are some examples of column matrices: $A = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}, \quad B = \begin{pmatrix} 6 \\ 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 \\ 3 \\ -1 \\ 2 \end{pmatrix}$ Note: column matrices can be thought of as column vectors, in fact we'll often treat them that way as we learn more about matrices.

Identity Matrix

An identity matrix is a square matrix whose diagonal entries are all equal to $$1$$ and all other entries are equal to $$0$$. Here are some identity matrices: $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ $I_4 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$ as we'll be seeing, when we multiply a matrix by an identity matrix it doesn't change its value (just like when we multiply a number by $$1$$).

Exercise

1. State the order (size) of each of the following matrices:
1. $$A = \begin{pmatrix} 2 & -3 \\ 0 & 1\end{pmatrix}$$
1. $$B = \begin{pmatrix} 1 & -5 \\ 4 & 6 \\ 0 & 8\end{pmatrix}$$

1. $$C = \begin{pmatrix} 4 & 7 & 0 & -3 \\ 5 & 9 & 2 & 1 \end{pmatrix}$$
1. $$D = \begin{pmatrix} 8 \end{pmatrix}$$

1. $$E = \begin{pmatrix} 4 \\ -2 \\ 6 \end{pmatrix}$$
1. $$F = \begin{pmatrix} 8 & 0 & 2 \\ -2 & 3 & 4 \\ 5 & 1 & 0 \\ 7 & 9 & 10 \end{pmatrix}$$

2. Given the matrix $$A = \begin{pmatrix} 2 & 3 & -5 \\ 0 & 1 & 4 \\ 6 & -8 & 9 \end{pmatrix}$$, state the value of each of the following entries:
1. $$a_{2,3}$$
1. $$a_{1,2}$$
1. $$a_{2,1}$$

1. $$a_{1,3}$$
1. $$a_{3,3}$$
1. $$a_{2,3}$$

3. For each of the matrices below:
1. State its order
2. State whether it is a square matrix, a column matrix, a row matrix or neither of those.
1. $$A = \begin{pmatrix}3 & 5 \\ 1 & -2 \end{pmatrix}$$
1. $$B = \begin{pmatrix}7 \\ 1 \\ -3 \end{pmatrix}$$

1. $$C = \begin{pmatrix}2 & 3 & 9 \\ 1 & 4 & 8 \end{pmatrix}$$
1. $$D = \begin{pmatrix} -4 & 5 \end{pmatrix}$$

1. $$E = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$$
1. $$F = \begin{pmatrix}2 & -1 & 5 \\ 1 & 7 & 9 \\ 8 & 0 & 3 \end{pmatrix}$$

Note: this exercise can be downloaded as a worksheet to practice with: