Back to: Mathematics – Linear Algebra
Definition of a Matrix
😊 Hello students! Today, we’re diving into one of the most fundamental — and powerful — tools in all of mathematics: the matrix. Think of a matrix as a “data organizer.” It’s not just a grid of numbers; it’s a way to encode systems, transformations, and relationships. Ready? Let’s go!
1. What is a Matrix?
A matrix is a rectangular arrangement of numbers (real or complex) into m horizontal rows and n vertical columns, enclosed by brackets such as [ ] or parentheses ( ).
We denote a matrix by a capital letter, such as A, and its individual entries by lowercase letters with subscripts, like aij, where:
- i = row number
- j = column number
For example, consider the following matrix:
This is called an m × n matrix (read as “m by n”) because it has m rows and n columns.
🤔 Think about this: If you have a 3×4 matrix, how many total entries does it contain? (Answer: 12!)
2. Order (or Size) of a Matrix
The order (or dimension) of a matrix tells us how many rows and columns it has. It is always written as rows × columns.
Example:
Let
Matrix B has 2 rows and 3 columns ⇒ order = 2 × 3.
The element in row 2, column 3 is written as \( b_{23} = 1 \).
😠 Common mistake! Remember: Rows come first! A 2×3 matrix is not the same as a 3×2 matrix.
3. General Notation
We often abbreviate a matrix using the notation:
This tells us that A is built from entries aij arranged in m rows and n columns.
4. Equality of Matrices
Two matrices A and B are equal (written \( A = B \)) if and only if:
- They have the same order, and
- All corresponding entries are equal: \( a_{ij} = b_{ij} \) for every i, j.
Example:
Solve for x and y:
- From (1,1): \( x + y = 8 \)
- From (2,2): \( x – y = 2 \)
Adding: \( 2x = 10 \Rightarrow x = 5 \), so \( y = 3 \).
5. Why Matrices Matter 🌟
Matrices are not just academic exercises — they are used everywhere:
- Physics: Representing systems of forces or transformations.
- Computer Graphics: Rotating, scaling, or translating images.
- Economics: Input-output models (e.g., Leontief models).
- Machine Learning: Data is stored as matrices; operations use linear algebra.
- Engineering: Solving circuit equations or structural loads.
😄 Fun fact: The word “matrix” comes from Latin, meaning “womb” — because it’s a structure that “gives birth” to solutions! 🤯
6. Special Terminology
| Term | Meaning |
|---|---|
| Entry / Element | A single number in the matrix (e.g., \( a_{23} \)). |
| Row | Horizontal line of entries (e.g., row 1 = [a₁₁, a₁₂, …, a₁ₙ]). |
| Column | Vertical line of entries. |
| Square Matrix | When m = n (same number of rows and columns). |
💡 Tip: Always check the order first when working with matrices. Many operations (like multiplication) depend on matching dimensions!
7. Quick Self-Check 🧠
- If \( C = (c_{ij})_{4 \times 5} \), how many entries does C have?
- Is the following true?
\[ \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 & 2 \end{bmatrix} \]
- What is \( a_{32} \) in the matrix
\[ A = \begin{bmatrix} 9 & -1 & 4 \\ 0 & 5 & 2 \\ 7 & \mathbf{3} & 1 \end{bmatrix} \]?
Answers: (1) 20, (2) No — different orders! (3) 3
Definition of a Matrix – Questions & Full Solutions
😊 Hello students! Below are all the questions from both documents that directly relate to the definition of a matrix. Each includes a detailed, step-by-step solution with explanations. Use this to check your understanding!
Q1. Suppose \( A \) is a \( 5 \times 7 \) matrix. Then:
- \( A \) has 7 rows. (True/False)
- \( a_{ij} \) is an element of \( A \) for \( i = 6 \) and \( j = 4 \). (True/False)
- For what values of \( i \) and \( j \) is \( a_{ij} \) an element of \( A \)?
Solution:
- False. A \( 5 \times 7 \) matrix has 5 rows and 7 columns. The format is always \( \text{rows} \times \text{columns} \).
- False. Row indices go from 1 to 5. Since \( i = 6 > 5 \), \( a_{64} \) does not exist.
- \( a_{ij} \) exists for \( 1 \leq i \leq 5 \) and \( 1 \leq j \leq 7 \).
Q2. Suppose \[ A = \begin{bmatrix} 4 & 7 & -8 \\ 5 & 1 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -1 & 5 \\ 7 & 8 & 6 \\ 5 & 3 & 2 \end{bmatrix} \]
- What is the order of \( A \) and \( B \)?
- Find \( a_{22} \), \( a_{13} \), \( b_{13} \), and \( b_{31} \).
Solution:
- \( A \) is \( 2 \times 3 \), \( B \) is \( 3 \times 3 \).
-
- \( a_{22} = 1 \) (row 2, column 2 of \( A \))
- \( a_{13} = -8 \) (row 1, column 3 of \( A \))
- \( b_{13} = 5 \) (row 1, column 3 of \( B \))
- \( b_{31} = 5 \) (row 3, column 1 of \( B \))
Q3. Form a \( 4 \times 5 \) matrix \( B = (b_{ij}) \) such that \( b_{ij} = i + j \).
Solution:
We construct \( B \) where entry in row \( i \), column \( j \) is \( i + j \):
Q4. Given the matrix equation \[ \begin{bmatrix} x + y & 6 \\ 3 & x – y \end{bmatrix} = \begin{bmatrix} 8 & 6 \\ 3 & 2 \end{bmatrix} \] find \( x \) and \( y \).
Solution:
By definition of matrix equality, corresponding entries must be equal:
- From (1,1): \( x + y = 8 \)
- From (2,2): \( x – y = 2 \)
Add the two equations: \( 2x = 10 \Rightarrow x = 5 \)
Substitute: \( 5 + y = 8 \Rightarrow y = 3 \)
✅ Final answer: \( x = 5,\ y = 3 \)
Q5. Find values of \( x, y, z, w \) satisfying: \[ \begin{bmatrix} x + y & x – z \\ 2x – y & w \end{bmatrix} = \begin{bmatrix} -1 & 3 \\ 5 & 0 \end{bmatrix} \]
Solution:
Match entries:
- \( x + y = -1 \) (1)
- \( x – z = 3 \) (2)
- \( 2x – y = 5 \) (3)
- \( w = 0 \) (4)
Add (1) and (3): \( (x + y) + (2x – y) = -1 + 5 \Rightarrow 3x = 4 \Rightarrow x = \frac{4}{3} \)
From (1): \( \frac{4}{3} + y = -1 \Rightarrow y = -\frac{7}{3} \)
From (2): \( \frac{4}{3} – z = 3 \Rightarrow z = \frac{4}{3} – 3 = -\frac{5}{3} \)
From (4): \( w = 0 \)
✅ Final answer: \[ x = \frac{4}{3},\quad y = -\frac{7}{3},\quad z = -\frac{5}{3},\quad w = 0 \]
Q6. Explain why \( [9] \ne \begin{bmatrix} 9 \end{bmatrix} \) is false. Or is it?
Solution:
This is a trick question! In most contexts in linear algebra:
- \( [9] \) and \( \begin{bmatrix} 9 \end{bmatrix} \) both denote the same \( 1 \times 1 \) matrix.
The difference is purely notational — brackets vs. parentheses. Since both represent a single-element matrix of order \( 1 \times 1 \), they are equal.
✅ So the statement “they are not equal” is **false**.
📌 Key Reminders:
- Matrix order = (number of rows) × (number of columns)
- Two matrices are equal only if they have the same order and all corresponding entries match.
- Index \( a_{ij} \): \( i \) = row, \( j \) = column.