Matrices
WE BEGAN the review of matrices by concentrating on the addition and subtraction of matrices. You are asked to note the following:
In the addition of matrices, corresponding values are added. In the subtraction of matrices, corresponding values are subtracted.
Corresponding values in equal matrices are equal.
In the multiplication of matrix by a constant, all values of the matrix are multiplied by the constant.
This latter example may be illustrated as follows:
EXAMPLE
NB: Please be reminded that multiplication by a half is identical to divide by two.
MULTIPLICATION OF TWO MATRICES
The matrix Ax refers to the Matrix A with order x y; that is the x y x matrix with x rows and y columns.
It is important that you consider their orders when multiplying two matrices. The orders are reviewed to determine: If multiplication is possible.
The order of the product (matrix).
Given the matrices Ax & B z, the product can be found since x y x x the number of columns of A is the same as the number of rows of B; that is, y in each case.
The order of the answer is x z x
It is important to follow this procedure, especially if you are not comfortable with the topic.
Having established that both matrices can be multiplied, let us attempt the following: ( ) Find the product of A = ( 2, 3) and B = -5
1
Using the approach indicated previously to consider the orders of both, then A1x2x B2x1
They can be multiplied since there are 2 columns and 2 rows, respectively. The order of the product is 1 x 1.
The product is found as follows: ( )
( 2, 3) -5 = ( 2 x -5 + 3 ? 1) = ( -10 + 3) = ( -7)
1
This forms the basis of matrix multiplication, where you multiply row by column. This is repeated to other rows and columns in matrices.
Now, let us attempt the following together.
EXAMPLE SOLUTION
From the above, X2 x Y2 has order 2 x 1 x 2 x 1 Directed numbers is very important in this problem in order to evaluate the negative sign appropriately. Please review. The product of two 2 x 2 matrices has order 2 x 2
SOLUTION
You are encouraged to practise as many examples of the multiplication of two 2 x 2 matrices as possible, as these provide the most challenge with respect to the topic.
The following is a typical exam-type problem.
SOLUTION
EXAMPLE HOMEWORK