In this topic, we shall discuss different types of matrices.

(i) Column matrix

A matrix is said to be a column matrix if it has only one column. For example,

is a column matrix of order 4 × 1.

In general, A = [a_{ij}] _{m × 1} is a column matrix of order m × 1.

(ii) Row matrix

A matrix is said to be a row matrix if it has only one row. For example,

is a row matrix.

In general, B = [b_{ij}] _{1 × n} is a row matrix of order 1 × n.

(iii) Square matrix

A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example,

is a square matrix of order 3.

In general, A = [a_{ij}] _{m × m} is a square matrix of order m.

NOTE:

If A = [a_{ij}] is a square matrix of order n, then elements (entries) a_{11}, a_{22}, ..., a_{nn} are said to constitute the diagonal, of the matrix A. Thus, if

Then the elements of the diagonal of A are 3, 3√2 , –1.

(iv) Diagonal matrix

A square matrix B = [b_{ij}] _{m × m} is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [b_{ij}] _{m × m} is said to be a diagonal matrix if b_{ij} = 0, when i ≠ j.

For example,

are diagonal matrices of order 1, 2, 3, respectively.

(v) Scalar matrix

A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [b_{ij}] _{n × n} is said to be a scalar matrix if

b_{ij} = 0, when i ≠ j

b_{ij} = k, when i = j, for some constant k.

For example,

are scalar matrices of order 1, 2 and 3, respectively.

(vi) Identity matrix

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, the square matrix A = [a_{ij}] _{n × n} is an identity matrix, if

We denote the identity matrix of order n by I_{n}. When order is clear from the context, we simply write it as I. For example,

are identity matrices of order 1, 2 and 3, respectively.

Observe that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.

(vii) Zero matrix

A matrix is said to be zero matrix or null matrix if all its elements are zero. For example,

are all zero matrices. We denote zero matrix by O. Its order will be clear from the context.

Equality of matrices

Definition Two matrices A = [a_{ij}] and B = [b_{ij}] are said to be equal if

(i) they are of the same order

(ii) each element of A is equal to the corresponding element of B, that is a_{ij} = b_{ij} for all i and j.

For example,

are equal matrices but

are not equal matrices. Symbolically, if two matrices A and B are equal, we write A = B.

If

then x = – 1.5, y = 0, z = 2, a = √6, b = 3, c = 2.

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