Matrices Basic Operations

Equality of Matrices
Two matrices are equal if they have the same size and their corresponding elements are equal
For example,
if we are given that
     
then we can conclude that

          


Addition and Subtraction
The sum of two matrices of the same size is a matrix with elements
that are the sums of the corresponding elements in the two given matrices

     See Example 1, page 733, of the textbook

Since the elements of matrices are numbers, real or complex, and the addition of such numbers
obeys the commutative and associative laws, it follows that the addition of matrices also obeys
the commutative and associative laws

     Commutative Law      A + B = B + A
      Associative Law      ( A + B ) + C = A + ( B + C )

A matrix with elements that are all 0's is called a zero matrix

The negative of a matrix M, denoted by – M, is a matrix with elements
that are the negatives of the elements in M
For example,
if

     
Then

     

If A and B are matrices of the same size, we define subtraction as

               

In other words, to subtract matrix B from matrix A, we subtract corresponding elements

     See Examples 2, page 734, and Example 4, page 735, of the textbook


Multiplication of a Matrix by a Number
The product of a number k and a matrix M, denoted by kM, is a matrix
formed by multiplying each element of M by k

     See Examples 3 - 4, pages 734 - 735, of the textbook


Matrix Product
Matrix multiplication is involved in the process of rewriting systems of linear equations
as matrix equations
We can then solve the system of linear equations by solving the
corresponding matrix equation


Definition of the Product of a Row Matrix and a Column Matrix.
The product of a 1 x n row matrix and an n x 1 column matrix
is a 1 x 1 matrix given by


       
                   1 x n                               n x 1


                                                                          
                                                                                                                       1 x 1

To be able to multiply two matrices, the number of elements in the row matrix
and the column matrix must be the same

     See Examples 5 – 6, page 737, of the textbook


Using this definition, we can extend the definition of
Matrix Product
If A is an m x p matrix and B is a p x n matrix, then the matrix product
A and B, denoted by AB, is an m x n matrix whose element in the i-th row
and j-th column is the number obtained from the product of the i-th row of A
and the j-th column of B
If the number of columns in A does not equal the number of rows in B,
then the matrix product AB is not defined

     See Examples 7 – 8, pages 739 – 741, of the textbook


CAUTION:
Matrix multiplication is not commutative
     See Examples 7C and 7D, page 739, of the textbook

The zero property does not hold for matrix multiplication
That is, a product AB maybe zero with neither A nor B being equal to zero
     See Example 7D, page 739, of the textbook


               top
               next  Inverse of a Square Matrix