- Matrix operations
- matrix equation
- multiplication by a scalar
- Zero-one matrices
- Algebraic properties of matrices
- zero matrix
- identity matrix for multiplication
- main diagonal
- additive inverse
- multiplicative inverse
- Inverse property
- procedure for finding
- Systems of equations
- inverse method for finding solution
- calculator usage
16.4 Essential Ideas
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 ???
|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] +  =  + [M]||[I][M] = [M][I] = [M]|
|Inverse||[M] + [M] = [M] + [M] =||[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].