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Mathematics and Statistics: Matrices

Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. A matrix having m rows and n columns is called matrix of order m x n.
Operations performed on rows or columns of a matrix are called Elementary Transformation on a Matrix. There are six operations (3 for rows and 3 for columns) which are known as elementary transformation:

  • Interchanging any two rows (or columns)
  • Multiplication of the element of any row (or column) by a non-zero scalar quantity.
  • Addition of constant multiple of the elements of any row (or column) to the corresponding element of any other row (or column).
Two matrices are said to be equivalent, if one is obtained from other by elementary transformation.

  • The minor of any element of determinant of a matrix A, is the determinant obtained by deleting the row and the column in which that element is lying.
  • The co-factor of an element Matrices is the minor of Matrices with proper sign.
  • Co-factor is usually denoted by the corresponding capital letters, i.e. the co-factor of Matrices is denoted by Matrices

Elementary Transformations:

  • Interchanging Matrices and Matrices row is denoted by Matrices.
  • Multiplication of each element of Matrices row by non-zero k is denoted by Matrices.
  • Addition to the elements of Matrices row, the corresponding elements of Matrices row multiplied by non-zero k is denoted by Matrices.

Elementary Transformations:

  • Interchanging Matrices and Matrices column is denoted by Matrices.
  • Multiplication of each element of Matrices column by non-zero k is denoted by Matrices.
  • Addition to the elements of Matrices column, multiplied by non-zero k is denoted by Matrices.

  • A rectangular matrix does not possess inverse matrix.
  • Inverse of every square matrix, if exists, is unique.
  • A square matrix is invertible if and only if it is non-singular.
  • The inverse of any non-singular matrix can be obtained by elementary row transformations or by elementary column transformations.
  • By doing one more elementary operations if we obtain all Matrices in one or more rows of left side matrix of Matrices, then Matrices does not exist.
  • If A is a square matrix with n-rows; it is said to be invertible if there exists an n-rowed matrix B such that AB = BA = In [where In is the unit matrix for order n]. B is called inverse of A or Reciprocal of A.
  • The inverse of the inverse is the original matrix itself Matrices.
  • If A is a non-singular matrix, then Matrices
  • The inverse of the transpose of a matrix is the transpose of its inverse. Matrices
  • If A and B are two invertible matrices of the same order then AB is also invertible and Matrices.
  • If A, B, C are square matrices of the same order 'n'; If A is a non-singular matrix, then
    • Left Cancellation Law: AB = AC Matrices B = C
    • Right Cancellation Law: BA = CA Matrices B = C
  • If A is a non-singular matrix such that A is symmetric, then Matrices is also symmetric.

If inverse of a matrix exists; it can be found by using either elementary row transformation or elementary column transformation.
Let us assume; A be a square matrix of order 'n'. Then Matrices (or Matrices)
Let a sequence of elementary row (column) operation reduce A on the left (right) side to In. The same elementary row (column) operation reduces In on the left (right) side to a matrix B. Then In = BA.
Multiplying both sides by Matrices we get Matrices.
Using Associative Law we get Matrices.
We know that Matrices. So Matrices.

  • The adjoint of a square matrix A is the transpose of the matrix of co-factors of matrix A.
  • Adjoint of A is written as adj. A
  • Adjoint of a diagonal matrix is a diagonal matrix.
  • If A and B are two square matrices of the same order, then Matrices.
  • Matrices; where A is a non-singular matrix.
  • Matrices; where A is a non-singular matrix.
  • The inverse of any non-singular square matrix can be obtained by using: Matrices

Let us consider a system of linear equations
Matrices
The matrix form of these equations is : Matrices
It is in the form of AX = B ; where matrix A is the prefactor of X.
To find X, we perform row operation on A to reduce A to I. We perform same row operation on B and let B get transformed to B1.
Here AX = B gets transformed into IX = B1 form here we can obtain values of X.
Let’s go back to original condition, where AX=B.
Let’s assume A is the matrix of three equations with three unknowns. Instead of reducing A to I by means of row operation, we sometimes reduce A to a matrix in which first column is having elements 1, 0, 0 in the order and there at least one of the c, d, e and f is zero.

In the equation (in above section) AX = B . Let Matrices; so that Matrices exists uniquely. Pre-multiplying both sides by Matrices we get Matrices
On simplification we get Matrices.

  • If Matrices, then system of equations is consistent and has a unique solution.
  • If Matrices, and (adj A) B = 0 ; Then the system of equations is consistent and has many solutions.
  • If Matrices, and (adj A) B Matrices 0 ; Then the system of equations is inconsistent (has no solutions).
  • If all the constants, b1, b2 .... bn; are all zero; then system of equation is said to be homogenous. Here AX = O; where O is a null matrix.
    • If Matrices; then its only solution X=0 is called the trivial solution.
    • If Matrices, then AX=O have a non-trivial solution. It will have infinite solutions.
Note: Solutions of a system of linear equations can also be obtained by Rank Method.

  • The square matrix A is defined to be singular if and only if Matrices.
  • The square matrix A is defined to be non-singular if and only if Matrices.
  • Determinant of matrix A is denoted by det. A or Matrices.
  • Matrices is always a symmetric matrix.
  • Matrices is always a skew – symmetric matrix.
  • Matrices is always a symmetric matrix.
  • Only a square matrix has a diagonal.
  • If the determinants of two square matrices are equal, then those two matrices are need not be equal.
  • If A is a square matrix, then the sum of all diagonal elements of A is called trace of A.
  • A square matrix Matrices is said to be symmetric if Matrices element is the same as Matrices element. i.e. Matrices.
  • A square matrix Matrices is said to be skew-symmetric if Matrices element is negative of its Matrices element. i.e. Matrices .
  • Product AB is defined if and only if: Number of columns of A = Number of rows of B.
  • Addition is defined only for matrices, which are of the same type.
  • A non-zero matrix A is said to be in Echelon form , if A satisfies the following conditions:
    • All the non-zero rows of A , if any precedes the zero rows.
    • The number of zeros preceding the first non-zero element in a row is less than the number of such zeros in the succeeding row.
    • The first non-zero element in a row is unity.
  • Number of non-zero rows of a matrix given in the Echelon form is its rank.
  • Addition is defined only for those matrices which are of the same type.
  • Zero Matrix O is the additive identity, which is unique.
  • A square matrix Matrices is an identity matrix if Matrices .
  • Adjoint of a matrix of order 2 x 2 can be found by interchanging the elements of leading diagonal and changing the sign of the other two elements which are not in leading diagonal.
  • Types of Matrices
    • Row Matrix: Consists of only one row
    • Column Matrix: Consists of only one column
    • Square Matrix: Number of rows = Number of Columns
    • Upper Triangular Matrix: Elements below principal diagonal are all zero
    • Lower Triangular Matrix: Elements above principal diagonal are all zero.
    • Diagonal Matrix: All non-diagonal elements are zero.
    • Scalar Matrix: Diagonal Matrix with all elements of diagonal are equal.
    • Unit Matrix (Identity Matrix) : Diagonal Matrix with all diagonal elements=1
    • Zero Matrix (Null Matrix): Every element is zero
    • Transpose of Matrix: Obtained by interchanging rows and columns.
  • If co-factor of each element of a Matrix is kept at corresponding places then Matrix obtained is known as co-factor matrix.
  • In real life matrices are used in Cryptography, Economics, Sociology, Modern Psychology and various applications in Industrial management.