#### description

Column matrix, Row matrix, Square matrix, Diagonal matrix, Scalar matrix, Identity matrix, Zero matrix

#### notes

**Column matrix:**

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

For example, A = `[(0), (sqrt 3),(-1),(1/2)]` is a column matrix of order 4 x 1.

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

**Row matrix:**

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

For example, B = `[-1/2,sqrt 5,2,3]_(1 xx 4)` is a row matrix.

In general, B =`[b_(ij)]_(1 xx n)` is a row matrix of order 1 x n .

**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 A = `[(3,-1,0),(3/2,3sqrt2,1),(4,3,-1)]` is a square matrix of order 3.

In general , A = `[a_(ij)]_(m xx m)` is a square matrix of order m .

**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 , A =[4] , B = `[(-1 ,0),(0,2)]` , C = `[(-1.1,0,0),(0,2,0),(0,0,3)],` are diagonal matrices of order 1,2,3,respectively.

**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

A = [3] , B =` [(-1,0),(0,-1)] `,

C = `[(sqrt3,0,0),(0,sqrt3,0),(0,0,sqrt3)]`

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

**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 `a_(ij)` =

\[ a_{ij} = \begin{cases} 1 & \quad \text{if } \text{ i = j}\\ 0 & \quad \text{if } \text{ i ≠ j} \end{cases} \]

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 [1] , `[(1,0),(0,1)]`,`[(1,0,0),(0,1,0),(0,0,1)]`

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.

**Zero matrix:**

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

For example, [0],`[(0,0),(0,0)] , [(0,0,0),(0,0,0)]` , [(0) ,(0)]

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