Section 16.4: Inverse Matrices

16.4 Outline

  1. Matrix operations
    1. matrix equation
    2. equality
    3. addition
    4. multiplication by a scalar
    5. subtraction
    6. multiplication
    7. noncomformable
  2. Zero-one matrices
  3. Algebraic properties of matrices
    1. zero matrix
    2. identity matrix for multiplication
    3. main diagonal
    4. additive inverse
    5. multiplicative inverse
    6. properties
      1. commutative
      2. associative
      3. identity
      4. inverse
      5. distributive
  4. Inverse property
    1. nonsingular
    2. procedure for finding
    3. Systems of equations
      1. inverse method for finding solution
      2. calculator usage


16.4 Essential Ideas

Matrix Operations

Equality  [M] = [N] if and only if matrices [M] and [N] are the same order and the corresponding entries are the same.
Addition [M] + [N] = [S] if and only if [M] and [N] are the same order and the entries of [S] are found by adding the corresponding entries of [M] and [N].
Multiplication by a scalar c[M] = [M] is the matrix in which each entry of [M] is multiplied by the scalar (real number) c.
Subtraction [M][N] = [D] if and only if [M] and [N] are the same order and the entries of [D] are found by subtracting the entries of [N] from the corresponding entries of [M].
Multiplication Let [M] be an m x r matrix and [N] an r x n matrix.  The product matrix  [M][N] = [P] is an m x n matrix.  The entry in the i th row and j th column of [M][N] is the sum of the products formed by multiplying each entry of the i th row of [M] by the corresponding entry in the j th column of [N].

Properties of Matrices ???

     Property  Addition Multiplication
Commutative[M] + [N] = [N] + [M][M][N] does not equal [N][M]
 Associative([M] + [N]) + [P] = [M] + ([N] + [P])([M][N])[P] = [M]([N][P])
Identity[M] + [0] = [0] + [M][I][M] = [M][I] = [M]
Inverse[M] + [M] = [M] + [M] =[0]  [M][M] = [M][M] =[I]
Distributive[M]([N] + [P]) = [M][N] + [M][P]
Distributive([N] + [P])[M] = [N][M] + [P][M]

Inverse of a Matrix
If [A] is a square matrix and if there exists a matrix [A]-1 such that [A]-1[A] = [A][A]-1 = [I] where [I] is the identity matrix for multiplication, then [A]-1 is called the inverse of [A] for multiplication.

Procedure for Finding the Inverse of a Matrix

To find the inverse of a square matrix A:
Step 1 : Augment [A] with [I]; that is write [A | I], where [I] is the identity matrix of the same order as [A].
Step 2:  Perform elementary row operations using Gauss-Jordan elimination to change the matrix [A] into the identity matrix [I], if possible.
Step 3:  If at any time you obtain all zeros in a row or column to the left of the dividing line,  then there will be no inverse.
Step 4:  If steps 1 and 2 can be performed, the result in the augmented part is the inverse of [A].