# what is matrix in math

## Undersatnd use of  matrix in maths

A rectangular array of symbols (which could be real or complex numbers) along rows and columns is called a matrix.

Thus a system of m × n symbols arranged in a rectangular formation along m rows and n columns and bounded by the brackets [.] is called an m by n matrix (which is written as m x n matrix).Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah.

Thus,

### Types of matrices

Row Matrix:

A matrix having a single row is called a row matrix.  e. g. [1 3 5 7]

Column Matrix:

A matrix having a single column is called a column matrix.  e.g. .

Square Matrix:

An m x n matrix A is said to be a square matrix if m = n i.e. number of rows = number of columns.

For example:  is a square matrix of order 3 × 3.

### Trace of a Matrix:

The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A i.e.  tr(A)

Thus if A = [aij]n×n

Then tr(A) = = a11 + a22 + ..... + ann

Example: Find the trace of the matrix A = .

Detail Explanation :    tr (A) = 1 + (–1) + 4 = 4.

### Diagonal Matrix:

A square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [aij]n×n to be a diagonal matrix, aij = 0, whenever i ≠ j.

is a diagonal matrix of order 3 × 3.

Scalar Matrix:

A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix.

For a square matrix A = [aij]n×n to be a scalar matrix aij = , where m ≠ 0.

is a scalar matrix.

### Unit Matrix or Identity Matrix:

A diagonal matrix of order n which has unity for all its diagonal elements, is called a unit matrix of order n and is denoted by In.

Thus a square matrix A = [aij]n×n is a unit matrix if aij =

#### Triangular Matrix:

A square matrix in which all the elements below the diagonal elements are zero is called Upper Triangular matrix and a square matrix in which all the elements above diagonal elements are zero is called Lower Triangular matrix.

Given a square matrix A = [aij]n×n,

For upper triangular matrix, aij = 0,     i > j

and for lower triangular matrix, aij = 0,  i < j

: are respectively upper and lower triangular matrices.

Null Matrix:

If all the elements of a matrix (square or rectangular) are zero, it is called a null or zero matrix.

For A = [aij] to be null matrix, aij = 0  ∀ i, j

is a zero matrix

#### Transpose of a Matrix:

The matrix obtained from any given matrix A, by interchanging rows and columns, is called the transpose of A and is denoted by A′.

If A = [aij]m×n and A′ = [bij]n×m then bij = aji, ∀ i, j

If   A = , then

## Properties of Transposes:

• (A′)′ = A.
• (A ± B)′ = A′ ± B′, A and B being conformable matrices.
• (αA)′ = αA′, α being scalar.
• (AB)′ = B′A′, A and B being conformable for multiplication.

Conjugate of a Matrix:

The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by .

.

## Properties of Conjugates:

•
•
• , α being any number
•    , A and B being conformable for multiplication

Transpose Conjugate of a Matrix:

The transpose of the conjugate of a matrix A is called transposed conjugate of A and is denoted by Aθ. The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e. = Aθ

If A = [aij]m×n, then Aθ = [bji]n×m  where bji =

i.e. the (j, i)th element of Aθ = the conjugate of (i, j)th element of A.

If A =   then Aθ =

## Properties of Transpose conjugate:

• (Aθ)θ = A
• (A + B)θ  = Aθ + Bθ
• (kA)θ = Aθ, k being any number
• (AB)θ = BθAθ