Operations on Matrices

The Concept of a Matrix
Explain the concept of a matrix
A matrix is an array or an Orderly arrangement of objects in rows and columns.
Each object in the matrix is called an element (entity).
Consider the following table showing the number of students in each stream in each form.
Stream A 38 35 40 28
Stream B 36 40 34 39
Stream C 40 37 36 35
From the above table, if we enclose the numbers in brackets without changing their arrangement, then a matrix is farmed, this can be done by removing the headings and the bracket enclosing the numbers (elements) and given a name (normally a capital letter).
Nowthe above information can be presented in a matrix form as
28 1444994221034
Any matrix has rows and columns but sometimes you may find a matrix with only row without Colum or only column without row.
In the matrix A above, the numbers 38, 36 an 40 form the first column and 38, 35, 40 and 28 form the first row.
Matrix A above has three (3) rows and four (4) columns.
In the matrix A, 34 is the element (entity) in the second row and third column while 28 lies in the first row and fourth column. The plural form of matrix is matrices.
Normallymatrices are named by capital letters and their elements by small letters which represent real numbers.
30 1445114527850
Order of a matrix (size of matrix)
The order of a matrix or size of a matrix is given by the number of its rows and the number of its columns.
So if A has m rows and n columns, then the order of matrix is m x n.
It is important to note that the order of any matrix is given by stating the number of its rows first and then the number of its columns.
31 1445159078899
Types of matrices:
The following are the common types of matrices:-
32 1445159476013
33 1445159511306
Matrices of order up to 2 X 2
Add matrices of order up to 2 X 2
When adding or subtracting one matrix from another, the corresponding elements (entities) are /added or subtracted respectively.
This being the case, we can only perform addition and subtraction of matrices with the same orders.
Example 1
Given that
34 1445160365343
35 1445160393025
Matrices of order up to 2 X 2
Subtract matrices of order up to 2 X 2
Example 2
Given that
36 1445160579084
37 1445160688121
Example 3
Solve for x, y and z in the following matrix equation;
38 1445161042598
Exercise 1
Determine the order of each of the following matrices;
39 1445161224845
2. Given that
40 1445161317504
3. Given that
41 1445161487012
4. A house wife makes the following purchases during one week: Monday 2kg of meat and loaf of bread Wednesday, 1kg of meat and Saturday, 1kg of meat and one loaf of bread. The prices are 6000/= per kg of meat and 500/= per loaf of bread on each purchasing day
  1. Write a 3×2 matrix of the quantities of items purchased over the three days .
  2. Write a 2×1 column matrix of the unit prices of meat and bread.
5. Solve for x, y and z in the equation
42 1445161663788
Additive identity matrix.
If M is any square matrix, that is a matrix with order mxm or nxn and Z is another matrix with the same order as m such that
M+ Z= Z+M = M then Z is the additive identity matrix.
43 1445162059730
The additive inverse of a matrix.
If A and B are any matrices with the same order such that A+B = Z, then it means that either A is an additive inverse of B or B is an additive inverse of A that is B=-A or A= -B
44 1445162167427
Example 4
Find the additive inverse of A,
45 1445162295139
Example 5
Find the additive identity of B if B is a 3×3 matrix.
46 1445162416518
A Matrix of Order 2 X 2 by a Scalar
Multiply a matrix of order 2 X 2 by a scalar
A matrix can be multiplied by a constant number (scalar) or by another matrix.
Scalar multiplication of matrices:
Rule: If A is a matrix with elements say a, b, c and d, or
48 1445162610993
Example 6
Given that
49 1445162769670
50 1445162880509
51 1445162905960
Example 7
52 1445163185607
53 1445163298072
Two Matrices of order up to 2 X 2
Multiply two matrices of order up to 2 X 2
Multiplication of Matrix by another matrix:
54 1445164409212
AB is the product of matrices A and B while BA is the product of matrix B and A.
55 1445164474481
In AB, matrix A is called a pre-multiplier because it comes first while matrix B is called the post multiplier because it comes after matrix A.
Rules of finding the product of matrices;
  1. The pre –multiplier matrix is divided row wise, that is it is divided according to its rows.
  2. The post multiplier is divided according to its columns.
  3. Multiplication is done by taking an element from the row and multiplied by an element from the column.
  4. In rule (iii) above, the left most element of the row is multiplied by the top most element of the column and the right most element from the row is multiplied by the bottom most element of the column and their sums are taken:
56 1445164640825
Therefore it can be concluded that matrix by matrix multiplication is only possible if the number of columns in the pre-multiplier is equal to the number of rows in the post multiplier.
Example 8
Given That;
57 1445164818899
58 1445164842578
From the above example it can be noted that AB≠BA, therefore matrix by matrix multiplication does not obey commutative property except when the multiplication involves and identity matrix i.e. AI=IA=A
Example 9
60 1445165048904
Example 10
Find C×D if
61 1445165147340
Product of a matrix and an identity matrix:
If A is any square matrix and I is an identity matrix with the same order as A, then AI=IA=A
Example 11
62 1445165362633
Exercise 2
1. Given that A= (3 4) and
63 1445166292434
2. If,
64 1445166395030
3.Using the matrices
68 1445166926952
4.Find the values of x and y if
67 1445166727204

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