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 = [a_ij]_n×n

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

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 = [a_ij]_n×n to be a diagonal matrix, a_ij = 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 = [a_ij]_n×n,

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

and for lower triangular matrix, a_ij = 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 = [a_ij] to be null matrix, a_ij = 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 = [a_ij]_m×n and A′ = [b_ij]_n×m then b_ij = a_ji, ∀ 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 = [a_ij]_m×n, then A^θ = [b_ji]n×m  where b_ji =

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

If A =   then A^θ =

Properties of Transpose conjugate:

• (A^θ)^θ = A 
• (A + B)^θ  = A^θ + B^θ
• (kA)^θ = A^θ, k being any number 
• (AB)^θ = B^θA^θ