Types of Matrices | Mathematics (Maths) Class 12 - JEE PDF Download

Matrix Types: Overview

Matrices are rectangular arrays of numbers arranged in rows and columns. They are classified into many types according to their size, position of non-zero elements, special entries, symmetry properties and algebraic relations. Common types used in school and undergraduate mathematics include row and column matrices, zero (null) matrices, singleton matrices, horizontal and vertical matrices, square, diagonal, scalar and identity (unit) matrices, triangular matrices, symmetric and skew-symmetric matrices, Hermitian and skew-Hermitian matrices, singular and non-singular matrices, and several special matrices such as idempotent, nilpotent, periodic and involutary matrices.

Types of Matrices: Explanations

Row Matrix

Definition: A matrix having only one row is called a row matrix.

If A = [a_ij]_m×n then A is a row matrix when m = 1. The order of a row matrix is 1 × n.

Example: A = [1 2 4 5] is a row matrix of order 1 × 4. Another example is P = [-4 -21 -17] of order 1 × 3.

Column Matrix

Definition: A matrix having only one column is called a column matrix.

If A = [a_ij]_m×n then A is a column matrix when n = 1. The order of a column matrix is m × 1.

Example: The matrix above is a column matrix of order 4 × 1.

In the above example, P is of order 3 × 1 and Q is of order 5 × 1.

Zero or Null Matrix

Definition: A matrix in which every element is zero is called a zero or null matrix. It is denoted by 0.

If A = [a_ij]_m×n and a_ij = 0 for all i and j, then A is the zero matrix.

The above is a zero matrix of order 2 × 3.

The above is a 3 × 2 null matrix.

The above is a 3 × 3 null matrix.

Singleton Matrix

Definition: A matrix with only one element is called a singleton matrix. It is a 1 × 1 matrix.

Examples: [2], [3], [a] are singleton matrices.

Horizontal and Vertical Matrices

Horizontal matrix: A matrix of order m × n is called horizontal if n > m.

Vertical matrix: A matrix of order m × n is called vertical if m > n.

Square Matrix

Definition: A matrix is called square if the number of rows equals the number of columns, i.e. m = n. The order of a square matrix is n × n.

The above is a square matrix of order 3 × 3.

Trace: The sum of the diagonal elements (principal diagonal) of a square matrix A is called the trace of A and is denoted by tr(A).

In the above, P is of order 2 × 2 and Q is of order 3 × 3.

Diagonal Matrix

Definition: A square matrix is a diagonal matrix if all its non-diagonal elements are zero. That is, for a square matrix A = [a_ij], a_ij = 0 whenever i ≠ j.

The above is a diagonal matrix of order 3 × 3; it may be written as diagonal[2, 3, 4].

Important points:

• A diagonal matrix is always a square matrix.
• Diagonal elements are a_ii, appearing where i = j.

More examples:

P = [9]

Here P, Q and R are diagonal matrices of orders 1 × 1, 2 × 2 and 3 × 3 respectively.

Scalar Matrix

Definition: A diagonal matrix in which all diagonal entries are equal is called a scalar matrix. Such a matrix can be written as λI, where λ is a scalar and I is the identity matrix of appropriate order.

]_mx is a scalar matrix if a_ij =  where k is a constant. 

is a scalar Matrix. 

If all diagonal elements equal 1 (that is λ = 1), the scalar matrix becomes the identity (unit) matrix.

Unit Matrix or Identity Matrix

Definition: A square diagonal matrix whose diagonal entries are all 1 is called the identity or unit matrix. An identity matrix of order n is denoted by I_n.

Properties:

• For any matrix A of order m × n, I_mA = A and AI_n = A whenever the multiplications are defined.
• All identity matrices are scalar matrices.
• All scalar matrices are diagonal matrices.
• All diagonal matrices are square matrices.

Note: The converses are not generally true; e.g., a square matrix need not be diagonal, a diagonal matrix need not be scalar, and a scalar matrix need not be the identity unless λ = 1.

Equal Matrices

Definition: Two matrices are said to be equal if they are of the same order and all corresponding elements are equal.

Conditions for equality of matrices A = [a_ij]_m×n and B = [b_ij]_r×s:

• m = r (same number of rows).
• n = s (same number of columns).
• a_ij = b_ij for all i = 1, 2, ..., m and j = 1, 2, ..., n (corresponding entries equal).

The matrices above are not equal because their orders differ.

If the matrices above are equal then their corresponding entries give relations such as a₁ = 1, a₂ = 6, a₃ = 3, b₁ = 5, b₂ = 2, b₃ = 1, etc., as illustrated.

Triangular Matrix

Definition: A square matrix is called a triangular matrix if all entries either above or below the principal diagonal are zero.

Upper Triangular Matrix

A square matrix [a_ij] is upper triangular if a_ij = 0 whenever i > j (all entries below the principal diagonal are zero).

The above is an upper triangular matrix of order 3 × 3.

Lower Triangular Matrix

A square matrix is lower triangular if a_ij = 0 whenever i < j (all entries above the principal diagonal are zero).

The above is a lower triangular matrix of order 3 × 3.

Singular and Non-Singular Matrices

Definition: For a square matrix A, if det(A) = 0 then A is called a singular matrix. If det(A) ≠ 0 then A is called a non-singular (or invertible) matrix.

Remarks:

• If A is non-singular, A has an inverse A^-1 and A A^-1 = I.
• If A is singular, A does not have a multiplicative inverse.

Symmetric and Skew-Symmetric Matrices

Symmetric matrix: A square matrix A = [a_ij] is symmetric if a_ij = a_ji for all i, j. Equivalently, A^T = A (where A^T denotes the transpose of A).

The matrix above is symmetric because a₁₂ = 2 = a₂₁, a₃₁ = 3 = a₁₃, etc.

Skew-symmetric matrix: A square matrix A = [a_ij] is skew-symmetric if a_ij = -a_ji for all i, j. Putting i = j gives a_ii = -a_ii, so a_ii = 0 for all diagonal entries.

Both matrices above are skew-symmetric. Note that for a skew-symmetric matrix A, A^T = -A.

Important conclusions:

• If A is any square matrix then A + A^T is symmetric and A - A^T is skew-symmetric.
• Every square matrix A can be written uniquely as the sum of a symmetric matrix and a skew-symmetric matrix: A = 1/2(A + A^T) + 1/2(A - A^T).
• If A and B are symmetric and they commute (AB = BA), then AB is symmetric.
• If B^T A B is formed, it is symmetric or skew-symmetric corresponding to A being symmetric or skew-symmetric respectively.
• All positive integral powers of a symmetric matrix are symmetric.
• Positive odd powers of a skew-symmetric matrix are skew-symmetric; positive even powers are symmetric.

Hermitian and Skew-Hermitian Matrices

Hermitian matrix: For matrices with complex entries, a square matrix A = [a_ij] is called Hermitian if a_ij = overline{a_ji} for all i, j, i.e. A^* = A where A^* denotes the conjugate transpose (also written Ā^T).

The matrices above are Hermitian.

Notes:

• If a Hermitian matrix has only real entries, it is a real symmetric matrix.
• For a Hermitian matrix, all diagonal elements are real.
• For a skew-Hermitian matrix A, A^* = -A; diagonal entries of a skew-Hermitian matrix are purely imaginary or zero.

If A is a Hermitian matrix then thus every diagonal element of a Hermitian Matrix must be real. 

A = [a_ij] is said to be a skew-Hermitian if  i.e. A^θ = – A; 

In particular, a skew-Hermitian matrix over the real numbers reduces to a real skew-symmetric matrix.

Special Matrices

Idempotent Matrix

Definition: A square matrix A is idempotent if A² = A.

For an idempotent matrix A, every eigenvalue is 0 or 1 and hence det(A) is 0 or 1 depending on the eigenvalues. In particular, idempotent matrices are diagonalizable (over ℝ or ℂ) with eigenvalues 0 or 1.

Nilpotent Matrix

Definition: A square matrix A is called nilpotent of index p (p ∈ ℕ) if A^p = 0 and A^p-1 ≠ 0. The least positive integer p with this property is the index of nilpotency.

Periodic Matrix

Definition: A square matrix A is periodic with period k if A^k+1 = A and k is the least positive integer with this property. If k = 1 then A is idempotent.

The matrix above has period 1 (hence idempotent).

Notes:

• The period of the zero matrix is not defined in this sense (A^k+1 = A would force 0 = 0 for any k but the notion of least positive k fails).
• The period of an idempotent matrix is 1.

Involutary Matrix

Definition: A square matrix A is called involutary if A² = I. Such a matrix is its own inverse.

Example: A matrix satisfying A² = I is involutary and A = A^-1.

Applications and Remarks

Matrices of specific types are used widely in algebra, geometry, physics, computer science and statistics. Typical uses include:

• Identity and diagonal matrices: simplify solutions of linear systems, help compute determinants and eigenvalues rapidly.
• Triangular matrices: appear in Gaussian elimination and LU decomposition; determinants equal the product of diagonal entries.
• Symmetric and Hermitian matrices: have real eigenvalues and orthogonal (or unitary) eigenvectors; important in quadratic forms and physics (observables in quantum mechanics are Hermitian).
• Nilpotent and idempotent matrices: arise in linear algebraic structure theory, projection operators and matrix exponentials.
• Singular versus non-singular: determine whether linear systems have unique solutions (non-singular) or infinitely many/none (singular).

Final Remarks

This chapter summarises the common classifications of matrices used in school and first-year undergraduate mathematics. For each type, remember the defining condition (pattern of zero/non-zero entries, algebraic relation such as A² = A, determinant conditions, or transpose/conjugate transpose relations), typical properties (determinant, trace, eigenvalue behaviour) and standard examples. Understanding these types and their properties is essential for solving problems in linear algebra, matrix algebra and their applications.