Any rectangular array of numbers is called matrix. A matrix of order m × n have m rows and n columns. The individual items in a matrix are called its elements.
Its element in the ith row and jth column is aij. We denote matrix by A, B, C, etc.
Types of Matrices
A 1 × n matrix is called a row matrix (row vector).
A m × 1 matrix is called column matrix (Column vector).
An n × n matrix is called a square matrix.
If in a square matrix aij = 0 whenever i ≠ j, then A is called a diagonal matrix.
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.
Zero (null) Matrix
A matrix with all elements zero is called zero (null) matrix. It is denoted by .
Operation on Matrices
Addition of Matrix
If A = [aij] and B = [bij] are two matrices of the same order, say m×n. Then, the sum of the two matrices A and B is defined as a matrix C = [cij] of order m×n , where cij = aij + bij, for all possible values of i and j.
Multiplication by Scalar
if A = [aij]m×n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.
Multiplication of Matrix
For multiplication of two matrices A and B, the number of columns in A should be equal to the number of rows in B. Let A = [aij] be an m×n matrix and B = [bjk] be an n×p matrix. Then the product of the matrices A and B is the matrix C of order m×p.
cik = ai1 b1k + ai2 b2k + ai3 b3k + ... + ain bnk
Ri ↔ Rj or Ci ↔ Cj
Ri → kRi or Ci → kCi
Ri → Ri + kRj or Ci → Ci + kCj
Transpose of a Matrix
If A = [aij] be an m×n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or (AT).
If A = [aij]m×n , then A′ = [aji]n×m
- (A′)′ = A
- (kA)′ = kA′
- (A + B)′ = A′ + B′
- (A B)′ = B′ A′
Symmetric & Skew Symmetric
A square matrix A = [aij] is said to be symmetric if A′ = A, that is, [aij] = [aji] for all possible values of i and j.
Skew Symmetric Matrix
A square matrix A = [aij] is said to be skew symmetric matrix if A′ = –A, that is aji = –aij for all possible values of i and j.
- For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew symmetric matrix.
- Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by A–1.
To every square matrix A = [aij] of order n, we can associate a number called determinant of the square matrix A. Determinant is denoted by |A| or det A or Δ. Only square matrices have determinants.
Determinant of a matrix of order one
Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a.
Determinant of a matrix of order two
det (A) = |A| = Δ = a11a22 – a21a12
Determinant of a matrix of order 3 × 3
Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and C3) giving the same value.
|A| = a11 (a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23) + a13 (a21 a32 – a31 a22)
- The value of the determinant remains unchanged if its rows and columns are interchanged. det (A) = det (A′)
- If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
- If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.
- If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
Area of a Triangle
The area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3)
Since area is a positive quantity, always take the absolute value of the determinant. The area of the triangle formed by three collinear points is zero.
Adjoint of Matrix
The adjoint of a square matrix A = [aij]n×n is defined as the transpose of the matrix [Aij]n×n, where Aij is the cofactor of the element aij. Adjoint of the matrix A is denoted by adj A.
For a square matrix of order 2, the adj A can be obtained by interchanging a11 and a22 and by changing signs of a12 and a21,
Cofactor of an element aij, denoted by Aij is defined by Aij = (–1)i+j Mij , where Mij is minor of aij.
Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.
A(adj A) = (adj A)A = |A|I
A square matrix A is said to be singular if |A| = 0
A square matrix A is said to be non-singular if |A| ≠ 0
If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
A square matrix A is invertible if and only if A is non-singular matrix.
Test of Consistency
Determinants and matrices van be used for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear equations.
A system of equations is said to be consistent if its solution (one or more) exists.
A system of equations is said to be inconsistent if its solution does not exist.
The system of equations can be written as, AX = B
Case I: If A is a non-singular matrix, then its inverse exists.
AX = B
A–1 (AX) = A–1 B (premultiplying by A–1)
(A–1A) X = A–1 B (by associative property)
I X = A–1 B
X = A–1 B
This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.
Case II: If A is a singular matrix, then |A| = 0.
In this case, calculate (adj A) B.
If (adj A) B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent.
If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.