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, what is matrix in math           



 

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. what is matrix in math.   

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:  what is matrix in math 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) = what is matrix in math = 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. 

what is matrix in math 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. 

what is matrix in math 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 =

  what is matrix in math

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

: what is matrix in mathare 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

  what is matrix in math 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 = what is matrix in math, then what is matrix in math

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 .

what is matrix in math. what is matrix in math

Properties of Conjugates:

  •  what is matrix in math
  •  what is matrix in math
  • what is matrix in math, α being any number
  •  what is matrix in math  , 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. what is matrix in math = 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 = what is matrix in math  then Aθ = what is matrix in math

Properties of Transpose conjugate:

  • (Aθ)θ = A 
  • (A + B)θ  = Aθ + Bθ
  • (kA)θ = what is matrix in math Aθ, k being any number 
  • (AB)θ = BθAθ

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