Chapter:3

What are Matrices ?

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.

  • The horizontal lines of elements are said to be, rows of the matrix.
  • The vertical lines of elements are said to be, columns of the matrix.

Order Of Matrix

A matrix having m rows and n columns is called a matrix of order m × n.

  • Namely A = [aij]m×n to indicate that A is a matrix of order m × n.

Equality Of Matrix

Two matrices A = [aij] and B = [bij] are said to be equal if :

  • They are of the same order.
  • Each element of A is equal to the corresponding element of B , that is aij = bij for all i and j .

Types Of Matrices

Row Matrix

A matrix is said to be a row matrix if it has only one row.

Square Matrix

A matrix in which the number of rows are equal to number of columns.

Diagonal Matrix

A square matrix is said to be a diagonal matrix if all its non-diagonal elements are zero.

Scalar Matrix

A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.

Zero Matrix

A matrix is said to be zero matrix or null matrix if all its elements are zero.

Identity Matrix

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.

Column Matrix

A matrix is said to be column matrix if it has only one column.

Transpose Of A Matrix

The transpose of a matrix is found by interchanging its row into columns or columns into rows. The transpose of matrix is represented by A' or AT.

Symmetric And Skew Symmetric Matrices

A symmetric and skew symmetric matrix both are square matrices. The symmetric matrix is equal to its transpose. The skew symmetric matrix is a matrix whose transpose is equal to its negative.

Operations On Matrices

There are certain operations on matrices, addition of matrices, multiplication of matrix by scalar and multiplication of matrices.

Addition of matrices

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]m × n , where cij = aij + bij for all possible values of i and j .

Multiplication of matrices

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of 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.

Multiplication of a matrix 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.

Properties

Properties of matrix addition

  • Commutative Law: If A = [aij] and B = [bij] are two matrices of the same order, say m × n, then A + B = B + A.
  • Associative Law: If A=[aij], B=[bij] and C=[cij] are three matrices of the same order, say m × n, then (A + B) + C = A + (B + C).
  • Additive Identity: Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A.
  • Additive Inverse: Let A = [aij]m×n be any matrix, then we have another matrix as -A = [-aij]m×n such that A + (-A) = (-A) + A = O.

Properties of multiplication of matrices

  • Associative Law: For any three matrices A,B and C. We have (AB)C = A(BC).
  • The Distribution Law: For any three matrices A,B and C. A(B+C) = AB + AC and (A+B)C = AC + BC
  • Existence of Multiplicative Identity: For every square matrix A, there exist an identity matrix of same order such that IA = AI = A.
  • Matrix multiplication is not commutative AB ≠BA

Properties of scalar multiplication of a matrix

  • k(A+B) = kA + kB, where k is scalar.
  • (k+l)A = kA + lA, where k and l are scalars.

Invertible Matrices

  • 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 denoted by A-1
  • Inverse of a square matrix, if it exists is unique.