Matrices and Determinants

Matrices and Determinants

The topic of a matrix was established by a scientist named James Sylvester in the 19th century. The arrangement of numbers in ‘m’ horizontally arranged rows and ‘n’ vertically arranged columns can be termed as a matrix. Different arithmetical operations can be performed on matrices. The number of rows and columns of matrices can be represented by its order which is in the form of subscripts. Few types of matrices include square, symmetric, row, column, triangular, skew-symmetric, diagonal, identity, scalar and singleton matrix.

Standard formulae of matrices are listed below:

Consider A and B to be square matrices of order n, and In is the unit matrix, then

(a) A (adjoint of matrix A) = | A | In = (adjoint of matrix A) A

(b) | adjoint of matrix  A | = | A |n-1 

(c) adjoint (adjoint of matrix  A) = | A |n-2 A

(e) |adjoint (adjoint of matrix  A)|=|A|(n-1)^2

(f) adjoint (AB) = (adjoint of matrix  B) (adjoint of matrix  A)

(g) adjoint (Am) = (adj A)m,

(h) adjoint (kA) = kn-1 (adjoint of matrix  A), k ∈ R

(i) adjoint (In) = In

(j) adjoint 0 = 0

(k) If matrix A is symmetric, then adjoint of matrix A is also symmetric

(l) If matrix A is diagonal, then the adjoint of matrix A is also diagonal.

(m) If matrix A is triangular, then the adjoint of matrix A is also triangular.

(n) If matrix A is singular, then | adjoint of matrix  A | = 0

A couple of applications of matrices are as follows:

  • Its applications can be mostly found in scientific fields. 
  • A linear transformation is one of the major applications of matrices.
  • Manipulating 3D models and representing them on a 2d screen can be done using matrices.
  • The set of probabilities in statistics can be explained through stochastic matrices.
  • The derivatives and exponential functions of higher dimensions can be generalised by matrix calculus.
  • For the study of economic relationships, matrices are used.

Determinants 

The concept of a determinant is the value that can be obtained from the elements of a square matrix. It is denoted as |A|. If the determinants are of order n, then it consists of n2 elements. If the determinant is 1, then the matrix is unimodular.

A few properties of determinants are given below.

  • The value of the determinant remains unchanged when the rows and columns are interchanged.
  • The sign of the determinant changes when any 2 rows or columns of the determinant are interchanged.
  • If a determinant consists of 2 same rows or columns, then its determinant is 0.
  • Say if a determinant is multiplied by a scalar value “k”, then its determinant value also gets multiplied by “k”.
  • If there exist 0s above or below the main diagonal of a determinant, then its determinant = product of the elements on the diagonal. 

Few applications of determinants include the following:

  • The determinant value can be utilised while finding the inverse of a matrix.
  • The concept of a determinant is useful while finding the solution for a linear system of equations using Cramer’s rule.
  • It can be employed in finding the estimation of points and in obtaining the interpolating polynomial that is appropriate.
  • Say if the row or column vectors of a matrix are said to be linearly dependent, then the determinant is 0.

The applications of determinants are vast. A few are listed above for reference.

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