What is a Matrix?
Definition: In our matrices lesson, a matrix is a rectangular array of numbers arranged in horizontal rows and vertical columns to effectively organize data.
Matrix Order Explained
Definition: If an arrangement has $m$ rows and $n$ columns, its dimensions are $m \times n$.
Matrix Order: $m \times n$Example: Let mathematical array $A$ have 2 rows and 3 columns.
The matrix order of $A$: $2 \times 3$Elements: The element in row $i$ and column $j$ is denoted by $a_{ij}$.
If matrix $A = \begin{pmatrix} 5 & -2 \\ 0 & 4 \end{pmatrix}$, then $a_{12} = -2$Example 1:
If $A = \begin{pmatrix} 3 & -2 & 7 \\ 5 & 1 & 0 \end{pmatrix}$, find:
- The order of matrix $A$.
- $a_{21}, a_{12}, a_{23}$
1. The order is $2 \times 3$.
2. $a_{21} = 5, \quad a_{12} = -2, \quad a_{23} = 0$.
Example 2:
If $A = (a_{ij})$ is a matrix of order $3 \times 2$ where $a_{ij} = \frac{i+j}{2}$, write the matrix $A$.
$a_{11} = \frac{1+1}{2} = 1, \quad a_{12} = \frac{1+2}{2} = 1.5$
$a_{21} = \frac{2+1}{2} = 1.5, \quad a_{22} = \frac{2+2}{2} = 2$
$a_{31} = \frac{3+1}{2} = 2, \quad a_{32} = \frac{3+2}{2} = 2.5$
Therefore, $A = \begin{pmatrix} 1 & 1.5 \\ 1.5 & 2 \\ 2 & 2.5 \end{pmatrix}$.
Types of Matrices
(1) Row Matrix: matrix has exactly one row ($1 \times n$).
Example: $A = \begin{pmatrix} 1 & 4 & -2 \end{pmatrix}$(2) Square Matrix: arrangement has number of rows equal to columns ($m = n$).
Example: $B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is $2 \times 2$.(3) Zero Matrix: structure has all elements equal to zero.
Symbol: $\mathcal{O}$. Example: $\mathcal{O}_{2 \times 2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$.(4) Identity Matrix: square array has 1 in main diagonal and 0 elsewhere.
Symbol: $I$. Example: $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.Transpose & Equality
Equality: $A = B$ if they have the same order and $a_{ij} = b_{ij}$ for all $i, j$.
Example 3:
If $\begin{pmatrix} x+8 & -5 \\ 3 & y-x \end{pmatrix} = \begin{pmatrix} 38 & -5 \\ 3 & 10 \end{pmatrix}$, find the values of $x$ and $y$.
$x + 8 = 38 \implies x = 30$
$y - x = 10 \implies y - 30 = 10 \implies y = 40$.
Transpose Matrix ($A^T$): interchanging rows and columns.
Example 4:
If $A = \begin{pmatrix} 2 & -1 \\ 3 & 0 \\ 5 & 4 \end{pmatrix}$, find $A^T$ and $(A^T)^T$.
$A^T = \begin{pmatrix} 2 & 3 & 5 \\ -1 & 0 & 4 \end{pmatrix}$
$(A^T)^T = \begin{pmatrix} 2 & -1 \\ 3 & 0 \\ 5 & 4 \end{pmatrix} = A$.
Example 5:
If $A = \begin{pmatrix} \cot 30^\circ & \sec 30^\circ \\ \csc 30^\circ & \sin 30^\circ \end{pmatrix}$, $B = \begin{pmatrix} \sqrt{3} & \frac{1}{2} y \\ \sqrt{3} x & \frac{1}{2} \end{pmatrix}$, and $A = B^t$, then find the value of each of: $x, y$.
$\because A = \begin{pmatrix} \sqrt{3} & \frac{2}{\sqrt{3}} \\ 2 & \frac{1}{2} \end{pmatrix}, \quad B^t = \begin{pmatrix} \sqrt{3} & \sqrt{3} x \\ \frac{1}{2} y & \frac{1}{2} \end{pmatrix}, \quad \because A = B^t$
$\therefore \sqrt{3} x = \frac{2}{\sqrt{3}} \quad \therefore x = \frac{2}{3}$
$\therefore \frac{1}{2} y = 2 \quad \therefore y = 4$
- $A$ is called a symmetric matrix if and only if $A = A^t$
- $A$ is called a skew symmetric matrix if and only if $A = -A^t$
Symmetric Matrix: A square matrix where $A = A^T$.
Skew-Symmetric Matrix: A square matrix where $A = -A^T$.
Mastery & Solutions
01 Pro Practice: Indexing
"If $A$ is a $2 \times 2$ matrix where $a_{ij} = i + j$, find $A$."
Answer:$A = \begin{pmatrix} 2 & 3 \\ 3 & 4 \end{pmatrix}$
02 Pro Practice: Symmetry
"If matrix $A = \begin{pmatrix} x & 5 \\ 5 & y \end{pmatrix}$ is a symmetric matrix, find $x$ and $y$."
Answer:Any values of $x, y \in \mathbb{R}$ satisfy the condition because $a_{12} = a_{21} = 5$.
Frequently Asked Questions (FAQ)
What is a matrix?
A matrix is a rectangular array of numbers or elements organized in rows and columns, primarily used in algebra to represent data or solve linear equations.
What is matrix order?
Matrix order (or dimension) describes the size of the matrix. It is written as $m \times n$, where $m$ is the number of horizontal rows and $n$ is the number of vertical columns.
What is transpose?
Transpose of a matrix is an operation that flips the matrix over its diagonal, switching its row and column indices. The transpose of matrix $A$ is denoted as $A^T$ or $A^t$.