Back to: Mathematics – Linear Algebra
Types of Matrices
😊 Hello students! In our journey through linear algebra, matrices come in many “flavors” — each with special structure and purpose. Understanding these types is crucial for solving systems, diagonalizing transformations, and even understanding quantum mechanics! Let’s explore them one by one — with care, depth, and clarity.
1. Square Matrix
A matrix is called a square matrix if the number of rows equals the number of columns. That is, an \( n \times n \) matrix is square.
Examples:
💡 Why it matters: Only square matrices can have determinants, inverses, eigenvalues, and concepts like trace.
2. Diagonal Matrix
A square matrix \( D = (d_{ij}) \) is called diagonal if all off-diagonal entries are zero: \[ d_{ij} = 0 \quad \text{whenever } i \ne j. \]
Notation: We often write \( D = \mathrm{diag}(5, -2, 3) \).
Properties:
- Multiplying diagonal matrices is trivial: just multiply corresponding diagonal entries.
- The determinant is the product of diagonal entries: \( \det(D) = 5 \cdot (-2) \cdot 3 = -30 \).
3. Scalar Matrix
A diagonal matrix in which all diagonal entries are equal is called a scalar matrix. It has the form \( kI \), where \( I \) is the identity matrix and \( k \in \mathbb{R} \).
4. Identity Matrix
The identity matrix \( I_n \) is a scalar matrix where the scalar is 1. It satisfies \( AI_n = I_nA = A \) for any \( n \times n \) matrix \( A \).
💡 It’s the “1” of matrix multiplication — the multiplicative identity.
5. Triangular Matrices
A square matrix is:
- Upper triangular if all entries below the main diagonal are zero (\( a_{ij} = 0 \) for \( i > j \)).
- Lower triangular if all entries above the main diagonal are zero (\( a_{ij} = 0 \) for \( i < j \)).
Upper triangular:
Lower triangular:
Key Fact: The determinant of a triangular matrix is the product of its diagonal entries.
For \( U \) above: \( \det(U) = 2 \cdot 3 \cdot 5 = 30 \).
6. Symmetric Matrix
A square matrix \( A \) is symmetric if it equals its transpose: \[ A^T = A \quad \text{or equivalently} \quad a_{ij} = a_{ji} \text{ for all } i, j. \]
💡 Symmetric matrices arise naturally in quadratic forms, covariance matrices in statistics, and energy functions in physics.
7. Skew-Symmetric (Anti-Symmetric) Matrix
A square matrix \( A \) is skew-symmetric if: \[ A^T = -A \quad \text{or} \quad a_{ij} = -a_{ji} \text{ for all } i, j. \] In particular, all diagonal entries must be zero (since \( a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0 \)).
Summary Table
| Type | Condition | Example Size |
|---|---|---|
| Square | \( m = n \) | \( 3 \times 3 \) |
| Diagonal | \( a_{ij} = 0 \) if \( i \ne j \) | \( \mathrm{diag}(1,2,3) \) |
| Scalar | \( kI \) | \( 5I_2 \) |
| Identity | \( I \), diagonal = 1 | \( I_4 \) |
| Upper Triangular | \( a_{ij} = 0 \) if \( i > j \) | See above \( U \) |
| Lower Triangular | \( a_{ij} = 0 \) if \( i < j \) | See above \( L \) |
| Symmetric | \( A^T = A \) | See symmetric example |
| Skew-Symmetric | \( A^T = -A \), diagonal = 0 | See skew example |
Self-Check: Are You Ready?
- Can a non-square matrix be symmetric?
- Is the zero matrix both symmetric and skew-symmetric?
- What is the transpose of a diagonal matrix?
- If \( A \) is symmetric and \( B \) is skew-symmetric, what is \( A + B \)?
Answers: (1) No, (2) Yes!, (3) Itself, (4) Neither — unless one is zero.
Types of Matrices – Questions & Full Solutions
😊 Hello students! Below are all the questions from both documents — plus extra challenging ones — related to types of matrices. Each includes a complete, step-by-step solution. Use this to test and deepen your understanding!
Q1. Classify each of the following matrices:
Solution:
- A is the identity matrix \( I_3 \) — diagonal entries = 1, off-diagonal = 0.
- B is a scalar matrix — all diagonal entries equal (2), off-diagonal = 0.
- C is upper triangular — all entries below main diagonal are 0.
- D is skew-symmetric — note \( D^T = -D \), and diagonal = 0.
Q2. Determine whether the matrix \[ M = \begin{bmatrix} 4 & -2 & 1 \\ -2 & 3 & 5 \\ 1 & 5 & 0 \end{bmatrix} \] is symmetric, skew-symmetric, or neither.
Solution:
Compute transpose:
Since \( M^T = M \), it is symmetric.
Q3. Construct a \( 3 \times 3 \) non-zero skew-symmetric matrix.
Solution:
Recall: \( A^T = -A \) and diagonal must be 0.
Example:
Check: \( A^T = -A \) ✓
Q4. Is every identity matrix a scalar matrix? Is every scalar matrix an identity matrix?
Solution:
- ✅ Yes — an identity matrix is a scalar matrix with scalar = 1.
- ❌ No — a scalar matrix like \( 5I \) is not an identity matrix unless the scalar is 1.
Q5. Give an example of a diagonal matrix that is not scalar.
Solution:
Diagonal ✓, but entries not all equal → not scalar.
Q6. (Challenging) Prove: If a matrix is both upper triangular and symmetric, then it must be diagonal.
Solution:
Let \( A \) be upper triangular → \( a_{ij} = 0 \) for \( i > j \).
Also, \( A \) symmetric → \( a_{ij} = a_{ji} \) for all \( i, j \).
For \( i < j \): since \( A \) is upper triangular, \( a_{ij} \) may be non-zero.
But by symmetry, \( a_{ji} = a_{ij} \). However, \( j > i \Rightarrow j > i \), so in lower part → but upper triangular forces \( a_{ji} = 0 \) when \( j > i \) (i.e., \( i < j \)).
Thus \( a_{ij} = a_{ji} = 0 \) for all \( i \ne j \).
✅ Hence, all off-diagonal entries are zero → \( A \) is diagonal.
Q7. Show that the only diagonal skew-symmetric matrix is the zero matrix.
Solution:
Let \( D = \mathrm{diag}(d_1, d_2, \dots, d_n) \) be skew-symmetric.
Then \( D^T = D = -D \) ⇒ \( D = -D \) ⇒ \( 2D = 0 \) ⇒ \( D = 0 \).
Alternatively: skew-symmetric ⇒ diagonal entries satisfy \( d_{ii} = -d_{ii} \) ⇒ \( d_{ii} = 0 \).
✅ So \( D = O \).
Q8. Can a non-zero matrix be both symmetric and skew-symmetric?
Solution:
Suppose \( A^T = A \) and \( A^T = -A \). Then \( A = -A \) ⇒ \( 2A = 0 \) ⇒ \( A = 0 \).
✅ So only the **zero matrix** satisfies both.
Q9. Show that any square matrix \( A \) can be written as \( A = S + K \), where \( S \) is symmetric and \( K \) is skew-symmetric.
Solution:
Define:
Then:
- \( S^T = \frac{1}{2}(A^T + A) = S \) → symmetric
- \( K^T = \frac{1}{2}(A^T – A) = -K \) → skew-symmetric
- \( S + K = \frac{1}{2}(A + A^T + A – A^T) = A \)
✅ Decomposition always exists and is unique.
Q10. Decompose \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \] into symmetric and skew-symmetric parts.
Solution:
Compute \( A^T = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \)
Symmetric part:
Skew-symmetric part:
Check: \( S + K = A \) ✓, \( S^T = S \), \( K^T = -K \) ✓
Q11. Is an upper triangular matrix with all diagonal entries equal to 1 necessarily the identity matrix?
Solution:
❌ No! Example:
Upper triangular, diagonal = 1, but not identity (off-diagonal ≠ 0).
Q12. To which types does the zero matrix belong?
Solution:
The \( n \times n \) zero matrix \( O \) is:
- Square ✓
- Diagonal ✓ (off-diagonal = 0)
- Scalar ✓ (all diagonal = 0, so \( 0 \cdot I \))
- Both symmetric and skew-symmetric ✓
- Both upper and lower triangular ✓
✅ It belongs to **all** these types (for square case).
📌 Key Takeaways:
- Identity ⇒ scalar ⇒ diagonal ⇒ square
- Triangular + symmetric ⇒ diagonal
- Only zero matrix is both symmetric and skew-symmetric
- Every square matrix = symmetric + skew-symmetric