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) = = a_{11} + a_{22} + ..... + 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^{θ}A^{θ}