MATRICES Operations with Matrices Properties of Matrix Operations The Inverse of a Matrix DEPARTMENT OF ECONOMICS PGGCG-11 ,CHANDIGARH

Operations with Matrices (i, j)-th entry : row: m column: n size: m×n

i-th row matrix row matrix j-th column matrix Square matrix: m = n

Diagonal matrix: Trace:

Example:

Equal matrix: Example: (Equal matrix)

Matrix addition: Example : (Matrix addition)

Scalar multiplication: Matrix subtraction: Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3A, (b) –B, (c) 3A – B

Sol: (a) (b) (c)

Matrix multiplication: Size of AB where

Example : Show that AB and BA are not equal for the matrices. and Sol: Note: Note:

Ex : (Find AB) Sol:

Properties of Matrix Operations Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix: Identity matrix of order n =

Properties of matrix addition and scalar multiplication: (1) A+B = B + A A + ( B + C ) = ( A + B ) + C (3) 1A = A (4) C ( A+B ) = cA + cB

Properties of zero matrices: Notes: 0m×n: the additive identity for the set of all m×n matrices –A: is the additive inverse of A

Transpose of a matrix:

Transpose of A matrix (a) (b) (c) Sol: (a) (b) (c) (c)

Properties of transposes:

Symmetric matrix: A square matrix A is symmetric if A = AT Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A Example: is symmetric, find a, b, c? Sol:

Ex: is a skew-symmetric, find a, b, c? Sol:

The Inverse of a Matrix (1) The inverse of A is denoted by The inverse of a matrix is unique. (1) The inverse of A is denoted by If A and B are invertible matrices of size n, then AB is invertible and