Mathematics > Matrices and Determinants

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Matrices

  • A matrix is an ordered rectangular array of numbers or functions.
  • A matrix having m rows and n columns is called a matrix of order m × n.
  •  is a column matrix.
  • is a row matrix.
  • An m × n matrix is a square matrix if m = n.
  • A =  is a diagonal matrix if  = 0, when i ≠ j.
  • A =  is a scalar matrix if   = 0, when i ≠ j,   = k, (k is some constant), when i = j.
  • A =  is an identity matrix, if  = 1, when i = j,  = 0, when i ≠ j.
  • A zero matrix has all its elements as zero.
  • A = = B if (i) A and B are of same order, (ii)  for all possible values of i and j.
  • – A = (–1)A
  • A – B = A + (–1) B
  • A + B = B + A
  • (A + B) + C = A + (B + C), where A, B and C are of same order.
  • k(A + B) = kA + kB, where A and B are of same order, k is constant.
  • (k + l ) A = kA + lA, where k and l are constant.
  • If A =  and , then  (i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC
  • If A = [aij]m × n, then A` or AT = (i) (A`)` = A, (ii) (kA)` = kA`, (iii) (A + B)` = A` + B`, (iv) (AB)` = B`A`
  • A is a symmetric matrix if A` = A.
  • A is a skew symmetric matrix if A` = –A.
  • Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.
  • Elementary operations of a matrix are as follows:  (i)  
  • If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A–1 and A is the inverse of B.
  • Inverse of a square matrix, if it exists, is unique.

Determinants

  • Determinant of a matrix A =  is given by .
  • For any square matrix A, the |A| satisfy following properties.
  • |A`| = |A|, where A` = transpose of A.
  • If we interchange any two rows (or columns), then sign of determinant changes.
  • If any two rows or any two columns are identical or proportional, then value of determinant is zero.
  • If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.
  • Multiplying a determinant by k means multiply elements of only one row (or one column) by k.
  • If  A =  , then k .A =k3 A
  • If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.
  • If to each element of a row or a column of a determinant the equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains same.
  • Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting ith row and jth column and denoted by.
  • Cofactor of of given by  = (– 1)i+ j
  • Value of determinant of a matrix A is obtained by sum of product of elements of a row (or a column) with corresponding cofactors. For example,
  • A =
  • If elements of one row (or column) are multiplied with cofactors of elements of any other row (or column), then their sum is zero. For example,  =0
  • A (adj A) = (adj A) A = |A| I, where A is square matrix of order n.
  • A square matrix A is said to be singular or non-singular according as |A| = 0 or |A| ≠ 0.
  • If AB = BA = I, where B is square matrix, then B is called inverse of A.
  • Also A–1 = B or B–1 = A and hence (A–1)–1 = A.
  • A square matrix A has inverse if and only if A is non-singular.
  • A-1 = 1/|A|
  • Unique solution of equation AX = B is given by X = A–1 B, where A ≠ 0.
  • A system of equation is consistent or inconsistent according as its solution exists or not.
  • For a square matrix A in matrix equation AX = B
  1. |A| ≠ 0, there exists unique solution\
  2. |A| = 0 and (adj A) B ≠ 0, then there exists no solution
  3. |A| = 0 and (adj A) B = 0, then system may or may not be consistent.

 Area of a triangle with vertices is given by

 

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