Types of Matrix

Types of Matrices — Definitions & Examples

A concise reference listing common matrix types with a short definition and two examples for each. Perfect for students, teachers, and blog readers.

Table of contents

• Row Matrix
• Column Matrix
• Square Matrix
• Rectangular Matrix
• Zero (Null) Matrix
• Identity Matrix
• Diagonal Matrix
• Scalar Matrix
• Triangular Matrices
• Symmetric Matrix
• Skew-Symmetric Matrix
• Singular Matrix
• Non-Singular Matrix
• Orthogonal Matrix
• Sparse Matrix

Row Matrix

Definition: A matrix with only one row.

Examples

[ 3  5  7 ]
[ -2  4  9  1 ]

Column Matrix

Definition: A matrix with only one column.

Examples

[ 4
 -1
  2 ]

[ 7
  0 ]

Square Matrix

Definition: A matrix with an equal number of rows and columns (n × n).

Examples

[ 1  2
  3  4 ]

[ 5  6  7
  1  0  2
  3  4  8 ]

Rectangular Matrix

Definition: A matrix where the number of rows and columns are not equal.

Examples

[ 1  2  3
  4  5  6 ]

[ 7  8
  9 10
 11 12 ]

Zero (Null) Matrix

Definition: A matrix in which every element is zero.

Examples

[ 0  0
  0  0 ]

[ 0  0  0 ]

Identity Matrix

Definition: A square matrix with 1's on the main diagonal and 0's elsewhere (I_n).

Examples

I2 = [ 1  0
       0  1 ]

I3 = [ 1  0  0
       0  1  0
       0  0  1 ]

Diagonal Matrix

Definition: A square matrix where all non-diagonal entries are zero (only diagonal may be non-zero).

Examples

[ 3  0  0
  0  5  0
  0  0  7 ]

[ 1  0
  0 -4 ]

Scalar Matrix

Definition: A diagonal matrix in which every diagonal element is the same scalar value.

Examples

[ 5  0
  0  5 ]

[ -3  0  0
   0 -3  0
   0  0 -3 ]

Triangular Matrices

Upper triangular: all entries below the main diagonal are zero. Lower triangular: all entries above the main diagonal are zero.

Upper triangular examples

[ 1  2  3
  0  5  6
  0  0  7 ]

[ 4 -2
  0  9 ]

Lower triangular examples

[ 2  0  0
  3  4  0
  5  6  7 ]

[ 1  0
 -3  8 ]

Symmetric Matrix

Definition: A square matrix equal to its transpose (A = A^T).

Examples

[ 2  3
  3  5 ]

[ 1  4  7
  4  2  6
  7  6  3 ]

Skew-Symmetric Matrix

Definition: A square matrix where A^T = -A. Diagonal elements must be zero.

Examples

[  0 -2
   2  0 ]

[  0  3 -1
  -3  0  5
   1 -5  0 ]

Singular Matrix

Definition: A square matrix with determinant equal to zero (det(A) = 0).

Examples

[ 1  2
  2  4 ]  

[ 3  6
  1  2 ]

Non-Singular Matrix

Definition: A square matrix with non-zero determinant (det(A) ≠ 0). It has an inverse.

Examples

[ 1  2
  3  4 ]

[ 2  5
  1  3 ]

Orthogonal Matrix

Definition: A square matrix A such that A^TA = I (transpose is inverse).

Examples

[ 1  0
  0  1 ]

[ 0  1
 -1  0 ]

Sparse Matrix

Definition: A matrix with mostly zero entries (only a few non-zero elements).

Examples

[ 0  0  5
  0  0  0
  2  0  0 ]

[ 0  3
  0  0
  0  0 ]

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