Matrices – Full Chapter Notes for Linear Algebra I (Ethiopian University)
Hello, dear students! 😊 Welcome to your complete guide on Chapter 3: Matrices from your Linear Algebra I course. Whether you’re preparing for a quiz, a mid-term, or your final exam, this page has everything you need: clear definitions, solved examples, practice questions with “Show Answer” buttons, and real-life connections. Let’s learn together—step by step! 🙌
You’ll also find helpful external resources from Khan Academy, Wikipedia, and our Ethiopian friends at Ethio Temari (check their Telegram, YouTube, and TikTok too!).
1. What Is a Matrix?
A matrix looks like a table of numbers. We use capital letters like A, B, or C to name them. If a matrix has m rows and n columns, we say it has order m × n (read as “m by n”).
Each number inside the matrix is called an entry or element. The entry in row i and column j is written as aij.
Here’s the general form of a matrix:
Imagine you are recording the scores of 3 students in 4 subjects. You could arrange the data in a 3 × 4 matrix: each row = a student, each column = a subject. This makes it easy to compare and analyze!
Question: The matrix \( A = \begin{bmatrix} 1 & 5 & 2 \\ 0 & 3 & 6 \end{bmatrix} \) has order:
- 3 × 2
- 2 × 3
- 6 × 1
- 1 × 6
Answer: B. 2 × 3
There are 2 rows and 3 columns. So order = 2 × 3. Remember: ROWS first, then COLUMNS! 😃
2. Equality of Matrices
So, even if two matrices look similar, if one has 2 rows and the other has 3, they are not equal. Also, if their sizes match but one number is different, they’re still not equal.
Example:
\[
\begin{bmatrix} 5 & 1 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 2+3 & 1 \\ 3 & 2 \times 1.5 \end{bmatrix}
\]
Both sides simplify to the same numbers, so they are equal.
But:
\[
\begin{bmatrix} 9 \\ 2 \end{bmatrix} \neq \begin{bmatrix} 9 & 2 \end{bmatrix}
\]
Why? One is 2 × 1 (column), the other is 1 × 2 (row). Different orders! ❌
Question: Find \( x \) and \( y \) if:
\[
\begin{bmatrix} x + y & 6 \\ x – y & 3 \end{bmatrix} = \begin{bmatrix} 8 & 6 \\ 2 & 3 \end{bmatrix}
\]
Answer:
Compare entries:
\( x + y = 8 \) and \( x – y = 2 \).
Add the equations: \( 2x = 10 \Rightarrow x = 5 \).
Then \( y = 8 – 5 = 3 \).
So \( x = 5 \), \( y = 3 \). ✅
3. Types of Matrices
3.1 Row and Column Matrices
- Row matrix: Only one row. Example: \( [5\ 2\ -1\ 4] \) → order 1 × 4.
- Column matrix: Only one column. Example: \( \begin{bmatrix} 1 \\ 3 \\ 4 \end{bmatrix} \) → order 3 × 1.
3.2 Zero (Null) Matrix
A matrix where all entries are 0. Written as \( O_{m \times n} \) or just \( 0 \).
Example: \( O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \) is a 2 × 3 zero matrix.
3.3 Square Matrix
A matrix with equal rows and columns (order n × n).
Example: \( \begin{bmatrix} -3 & 4 & 6 \\ 2 & 1 & 3 \\ 5 & 2 & -1 \end{bmatrix} \) is a 3 × 3 square matrix.
3.4 Main Diagonal and Trace
In a square matrix, the main diagonal runs from top-left to bottom-right: \( a_{11}, a_{22}, \dots, a_{nn} \).
The trace is the sum of these diagonal entries.
Example:
\[
C = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 6 & 0 \\ 5 & 1 & 8 \end{bmatrix}
\quad \Rightarrow \text{Trace}(C) = 3 + 6 + 8 = 17
\]
3.5 Triangular Matrices
- Upper triangular: All entries below the main diagonal are 0.
Example: \( \begin{bmatrix} 2 & 4 & 8 \\ 0 & 1 & 2 \\ 0 & 0 & -3 \end{bmatrix} \) - Lower triangular: All entries above the main diagonal are 0.
Example: \( \begin{bmatrix} 5 & 0 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 6 & 1 & 2 & 0 \\ -2 & -4 & 8 & 6 \end{bmatrix} \)
3.6 Diagonal, Scalar, and Identity Matrices
Diagonal matrix: All off-diagonal entries are 0.
Written as \( \text{diag}(a_{11}, a_{22}, \dots, a_{nn}) \).
Example: \( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -1 \end{bmatrix} \)
Scalar matrix: A diagonal matrix where all diagonal entries are the same.
Example: \( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \)
Identity matrix (\( I_n \)): A scalar matrix with 1s on the diagonal.
Example:
\[
I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad
I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
\]
This matrix is super important—it acts like “1” in multiplication!
Question: Which of the following is a scalar matrix?
- \( \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix} \)
- \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
- \( \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix} \)
- \( \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \)
Answer: C
Only option C has equal non-zero diagonal entries and zeros elsewhere. That’s the definition of a scalar matrix! 😃
4. Matrix Operations
4.1 Matrix Addition and Subtraction
You can add or subtract two matrices ONLY if they have the same order. You do it entry by entry.
Example:
\[
A = \begin{bmatrix} 30 & 18 \\ 12 & 10 \end{bmatrix}, \quad
B = \begin{bmatrix} 36 & 20 \\ 24 & 14 \end{bmatrix}
\]
Then:
\[
A + B = \begin{bmatrix} 66 & 38 \\ 36 & 24 \end{bmatrix}
\]
The negative of a matrix flips the sign of every entry.
If \( A = \begin{bmatrix} 2 & -1 & 5 \\ 4 & -6 & 0 \end{bmatrix} \), then:
\[
-A = \begin{bmatrix} -2 & 1 & -5 \\ -4 & 6 & 0 \end{bmatrix}
\]
4.2 Properties of Matrix Addition
- Commutative: \( A + B = B + A \)
- Associative: \( (A + B) + C = A + (B + C) \)
- Additive identity: \( A + 0 = A \)
- Additive inverse: \( A + (-A) = 0 \)
4.3 Scalar Multiplication
Multiply every entry by a number (scalar).
If \( A = \begin{bmatrix} 0 & 2 & 3 \\ 2 & 1 & 4 \end{bmatrix} \), then:
\[
\frac{1}{2}A + 3B = \frac{1}{2} \begin{bmatrix} 0 & 2 & 3 \\ 2 & 1 & 4 \end{bmatrix} + 3 \begin{bmatrix} 7 & 6 & 3 \\ 1 & 4 & 5 \end{bmatrix} = \dots
\]
(You’d compute each entry separately.)
4.4 Matrix Multiplication
This is trickier! You can multiply \( A \times B \) only if the number of columns in \( A \) equals the number of rows in \( B \).
If \( A \) is \( m \times n \) and \( B \) is \( n \times p \), then \( AB \) is \( m \times p \).
How to multiply:
The entry in row i, column j of \( AB \) is the dot product of row i of \( A \) and column j of \( B \).
Example:
\[
A = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}, \quad
B = \begin{bmatrix} 5 & 3 \\ 6 & 4 \\ 7 & 5 \end{bmatrix}
\]
Then:
\[
AB = \begin{bmatrix}
(2)(5)+(3)(6)+(4)(7) & (2)(3)+(3)(4)+(4)(5)
\end{bmatrix} = \begin{bmatrix} 56 & 38 \end{bmatrix}
\]
Suppose matrix \( A \) shows how many hours 2 workers spend on 3 tasks, and matrix \( B \) shows the hourly pay for each task. Then \( AB \) gives the total pay for each worker. But \( BA \) would be meaningless—tasks don’t “work” on workers! 😅
Question: If \( A \) is 3 × 5 and \( AB \) is 3 × 7, what is the order of \( B \)?
Answer: 5 × 7
Since \( A \) is 3 × 5, \( B \) must have 5 rows to match. The product is 3 × 7, so \( B \) has 7 columns. Hence, order = 5 × 7. ✅
5. Transpose of a Matrix
The transpose of matrix \( A \), written \( A^T \), is formed by swapping rows and columns.
Example:
\[
A = \begin{bmatrix} 2 & -4 & 6 \\ 3 & 1 & 4 \end{bmatrix}
\quad \Rightarrow \quad
A^T = \begin{bmatrix} 2 & 3 \\ -4 & 1 \\ 6 & 4 \end{bmatrix}
\]
Properties of Transpose
- \( (A + B)^T = A^T + B^T \)
- \( (kA)^T = kA^T \)
- \( (AB)^T = B^T A^T \) ← Note the order reversal!
- \( (A^T)^T = A \)
Symmetric and Skew-Symmetric Matrices
- Symmetric: \( A^T = A \) → \( a_{ij} = a_{ji} \)
Example: \( \begin{bmatrix} 2 & 1 & 5 \\ 1 & 0 & -3 \\ 5 & -3 & 6 \end{bmatrix} \) - Skew-symmetric: \( A^T = -A \) → \( a_{ij} = -a_{ji} \)
Note: All diagonal entries must be 0!
Example: \( \begin{bmatrix} 0 & 5 & 7 \\ -5 & 0 & 3 \\ -7 & -3 & 0 \end{bmatrix} \)
Question: Is every symmetric matrix square?
Answer: Yes!
Because \( A^T \) has order \( n \times m \) if \( A \) is \( m \times n \). For \( A = A^T \), we must have \( m = n \). So yes—symmetric matrices are always square. 😮
6. Elementary Row Operations and Row Echelon Form
To solve systems of equations, we use elementary row operations:
- Replacement: Add a multiple of one row to another: \( R_i \to R_i + k R_j \)
- Interchange: Swap two rows: \( R_i \leftrightarrow R_j \)
- Scaling: Multiply a row by a non-zero constant: \( R_i \to k R_i \) (k ≠ 0)
Row Echelon Form (REF)
A matrix is in REF if:
- All non-zero rows are above zero rows.
- The first non-zero entry (called a pivot) of a row is to the right of the pivot above it.
- All entries below a pivot are 0.
Reduced Row Echelon Form (RREF)
In addition to REF, RREF requires:
- Each pivot is 1.
- Each pivot is the only non-zero entry in its column.
Example of RREF:
\[
\begin{bmatrix}
1 & 0 & 0 & 29 \\
0 & 1 & 0 & 16 \\
0 & 0 & 1 & 1
\end{bmatrix}
\]
7. Rank of a Matrix
The rank of a matrix is the number of non-zero rows in its REF or RREF.
Example:
\[
\begin{bmatrix}
1 & 0 & 1 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}
\quad \Rightarrow \quad \text{Rank} = 2
\]
Question: What is the rank of the zero matrix of size 3 × 3?
Answer: 0
All rows are zero → no non-zero rows → rank = 0. Simple! 😃
8. Systems of Linear Equations and Matrices
A system like:
\[
\begin{aligned}
a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n &= b_1 \\
a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n &= b_2 \\
\vdots \\
a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n &= b_m
\end{aligned}
\]
can be written as:
\[
A X = B
\]
where:
\[
A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix}, \quad
X = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}, \quad
B = \begin{bmatrix} b_1 \\ \vdots \\ b_m \end{bmatrix}
\]
The augmented matrix is \( [A \,|\, B] \)—just stick \( B \) as an extra column to \( A \).
Consistency and Rank
If ranks are equal and equal to number of variables → unique solution.
If ranks are equal but less than number of variables → infinitely many solutions.
If ranks differ → no solution (inconsistent). 😠
9. Practice Activities with Full Solutions
Activity 1: Solve for Variables
Find \( x, y, z, w \) that satisfy:
\[
\begin{bmatrix} x – y & 2x + z \\ 2x – y & 3z + w \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}
\]
Solution:
Equate entries:
(1) \( x – y = -1 \)
(2) \( 2x + z = 5 \)
(3) \( 2x – y = 0 \)
(4) \( 3z + w = 13 \)
From (1) and (3):
Subtract (1) from (3): \( (2x – y) – (x – y) = 0 – (-1) \Rightarrow x = 1 \)
Then from (1): \( 1 – y = -1 \Rightarrow y = 2 \)
From (2): \( 2(1) + z = 5 \Rightarrow z = 3 \)
From (4): \( 3(3) + w = 13 \Rightarrow w = 4 \)
✅ Final answer: \( x=1, y=2, z=3, w=4 \)
Activity 2: Matrix Multiplication
Let \( R = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 5 \\ 3 & 1 & 1 \end{bmatrix} \), \( S = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 3 & 0 \\ 4 & 2 & 1 \end{bmatrix} \), \( T = \begin{bmatrix} 2 & 3 & 0 \\ -3 & 2 & 2 \\ -1 & 0 & 1 \end{bmatrix} \)
Find \( 2RS – 3ST \).
Note: This is a long computation! You’d first compute \( RS \) and \( ST \), then scale and subtract. We’ll skip full arithmetic here, but the key is:
– Confirm dimensions: all are 3×3, so multiplication is valid.
– Use dot products carefully.
– Always check your work step by step—matrix multiplication is easy to mess up! 😅
💡 Tip: Practice with smaller matrices first. Use Khan Academy for interactive drills!
Activity 3: Find RREF and Rank
Find the RREF and rank of:
\[
A = \begin{bmatrix}
1 & 3 & 5 & 7 \\
2 & 4 & 6 & 8 \\
3 & 5 & 7 & 9
\end{bmatrix}
\]
Solution:
Step 1: \( R_2 \to R_2 – 2R_1 \), \( R_3 \to R_3 – 3R_1 \)
→ \( \begin{bmatrix} 1 & 3 & 5 & 7 \\ 0 & -2 & -4 & -6 \\ 0 & -4 & -8 & -12 \end{bmatrix} \)
Step 2: \( R_2 \to (-1/2)R_2 \)
→ \( \begin{bmatrix} 1 & 3 & 5 & 7 \\ 0 & 1 & 2 & 3 \\ 0 & -4 & -8 & -12 \end{bmatrix} \)
Step 3: \( R_1 \to R_1 – 3R_2 \), \( R_3 \to R_3 + 4R_2 \)
→ \( \begin{bmatrix} 1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \)
This is RREF! Non-zero rows = 2 → Rank = 2. ✅
Final Words of Encouragement
Wow! You’ve just covered the entire chapter on Matrices—definitions, types, operations, transpose, rank, and linear systems. 🎉 Give yourself a pat on the back!
Remember: mastering matrices is like learning the alphabet of higher math. Once you’re comfortable here, vector spaces, determinants, and eigenvalues will feel much easier.
Keep practicing! Try more problems from your textbook or visit Ethio Temari for Ethiopian-focused support. You’ve got this! 💪
— Your friendly Linear Algebra teacher 😊