Algebra of Matrices

Addition of Matrices:

  • Definition: Matrices can be added if they have the same dimensions (same number of rows and columns).
  • Representation: If Aand B are m×n matrices, their sum C=A+B is obtained by adding corresponding elements.
  • Example:
    A=[1234],B=[5678]
    A+B=[1+52+63+74+8]=[681012]

Properties:

  1. Commutative: A+B=B+A if A and B have the same dimensions.

  2. Associative: (A+B)+C=A+(B+C) if all matrices involved have the same dimensions.

  3. Zero Matrix: A+O=A, where O is the zero matrix. Adding a zero matrix to any matrix yields the original matrix.

  4. Additive Inverse: For any matrix A, there exists a matrix A such that A+(A)=O, where O is the zero matrix.

Subtraction of Matrices:

Matrices can be subtracted from each other if they have the same dimensions. Subtraction is performed element-wise, subtracting the corresponding elements from one matrix with the corresponding elements from another matrix.

Representation:

Given two matrices A and B of the same dimensions:

  • The subtraction of A from B is denoted as BA=C, where Cij=BijAij.

Example:

Consider two matrices A and B as follows:

A=[3254]
B=[7169]

Their subtraction BA would be:

BA=[7169][3254]=[73126594]=[4115]

Properties:

  1. Commutative: AB=BA if Aand B have the same dimensions.

  2. Associative: (AB)C=A(BC) if all matrices involved have the same dimensions.

  3. Zero Matrix: AO=A, where Ois the zero matrix. Subtracting a zero matrix from any matrix yields the original matrix.

  4. Additive Inverse: For any matrix A, there exists a matrix A such that A+(A)=O, where O is the zero matrix.

Scalar Multiplication:

  • Definition: Multiplying a matrix by a scalar multiplies each element of the matrix by that scalar.
  • Representation: If k is a scalar and A is an m×n matrix, kA is obtained by multiplying each element of A by k.
  • Example:
    k=3,A=[2413]
    kA=3[2413]=[61239]

Properties:

  1. Associative Property: (kl)A=k(lA), where k and l are scalars and A is a matrix.

  2. Distributive Property: k(A+B)=kA+kB, where A and B are matrices and k is a scalar.

  3. Distributive Property with Scalars: (k+l)A=kA+lA, where k and l are scalars and A is a matrix.

Multiplication of Matrices:

  • Definition: The number of columns in the first matrix must be equal to the number of rows in the second matrix.
  • Representation: If Ais an m×n matrix and B is an n×p matrix, their product C=AB will be an m×p matrix.
  • Example:
    A=[1234],B=[5678]
    AB=[(1×5+2×7)(1×6+2×8)(3×5+4×7)(3×6+4×8)]=[19224350]
  1. Dimensions Compatibility Rule:

    • For matrix multiplication C=AB, the number of columns in matrix A must be equal to the number of rows in matrix B.
    • If A is an m×n matrix and B is an n×p matrix, the resulting matrix C=AB will be an m×p matrix.
  2. Element-wise Calculation:

    • To compute the element cij in the resulting matrix C=AB, the ith row of matrix A is multiplied element-wise with the jth column of matrix B, and the products are summed.
    • cij=ai1×b1j+ai2×b2j++ain×bnj.
  3. Non-Commutative Property:

    • Matrix multiplication is not commutative in most cases: ABBAThe order matters in matrix multiplication.

Properties of Matrix Algebra:

  • Associative Property: (AB)C=A(BC)
  • Distributive Property: A(B+C)=AB+AC
  • Not Commutative: In general, ABBA(matrix multiplication is not commutative).
  • Identity Matrix:
    • IA=AI=A - Multiplying a matrix by an identity matrix leaves the matrix unchanged.

    Inverse Matrix:

    • A×A1=A1×A=I - The inverse of a matrix, if it exists, multiplied by the original matrix results in the identity matrix.