Understanding and Using Matrix Algebra

Understanding and Using Matrix Algebra

Introduction to Matrix Algebra

  • A matrix is a rectangular array of numbers arranged in rows and columns.
  • The order of a matrix is identified as the number of rows (m) by the number of columns (n), written as m x n.
  • A matrix of order 1 x n or m x 1 is a vector. If n=m, the matrix is categorised as a square matrix.
  • If a square matrix has the same elements on its leading diagonal and zero elsewhere, it is known as an identity matrix.
  • A matrix created by switching the rows and columns of a given matrix is called its transpose.
  • The determinant is a special number that can be calculated from a square matrix.

Operations with Matrices

  • Matrix Addition and Subtraction: Two matrices can be added or subtracted if and only if their orders are same. They are combined element by element.
  • Scalar Multiplication: A matrix can be multiplied by a scalar by multiplying each element of the matrix with the scalar.
  • Matrix Multiplication: Two matrices A of order m x n and B of order n x p can be multiplied. The result is a matrix C of order m x p where each element is the sum of the products of the corresponding elements in the rows of A and in the columns of B.

Properties of Matrix Operations

  • Commutative Law: For addition, A + B = B + A but for multiplication, AB ≠ BA.
  • Associative Law: For both addition and multiplication, (A + B) + C = A + (B + C); (AB)C = A(BC)
  • Distributive Law: A(B + C) = AB + AC and (B + C)D = BD + CD
  • Multiplication by Zero and One: For any matrix A, 0 . A = A . 0 = 0 and I . A = A . I = A, where 0 is the zero matrix and I is the identity matrix.

Inverse of a Matrix

  • If for a square matrix A, there exists another matrix B such that AB = BA = I, then B is called the inverse of A, denoted as A⁻¹.
  • An inverse only exists if the determinant of the matrix is not equal to zero. If the determinant is zero, the matrix is called singular, or noninvertible.

Application of Matrices

  • Matrices are used in many fields from computer graphics to quantum physics, and they provide a convenient way to model linear transformations such as rotations, translations, and scaling in physics.

Note

Always remember the rules and anything in bold as they are essential in understanding and using matrix algebra.