MATRICES
Operations on matrices
A matrix represents another way of writing information. Here the information is written as rectangular array. For example two students Juma and Anna sit a math Exam and an English Exam. Juma scores 92% and 85%, while Anna scores 66% and 86%. This can be written as.
A size of a matrix is known as its order and is denoted by the number of rows times the number of columns. Therefore the order of above matrix is
each of the numbers in the matrix is called an element.Types of matrix
(i) Row matrix. This is a matrix having only one row. Thus
is a row matrix.(ii) Column matrix. This is a matrix having only one column. Thus
is a column matrix.(iii) Null matrix. This is a matrix with all its elements zero. Thus
is a null matrix or zero matrix.(iv) Square matrix. This is a matrix having the same number of rows and column. Thus
is a square matrix.(v) Diagonal matrix. This is a square matrix in which all the elements are zero except the diagonal elements. Thus
is a diagonal matrix. Note that: The diagonal in a matrix always runs from up left to lower right.
(vi) Unit matrix of identity matrix. This is a diagonal or square matrix in which the diagonal elements equal to 1. An identity matrix is usually denoted by the symbol I. Thus
I =
(vii) Equal matrix. Two matrices are said to be equal if they are of the some order, responding elements are equal.
An m x n matrix (E.g. matrix A) is a rectangular array of m x n real (or complex numbers) arranged in M horizontal rows and n vertical columns.
A =
Example
1. The table below represents number of students in each stream in each form. Now write that information in matrix.
Form | I | II | III | IV |
Stream J | 36 | 38 | 30 | 41 |
Stream K | 38 | 41 | 29 | 30 |
Stream L | 29 | 50 | 35 | 42 |
J =
2. Give the order of the following matrices.
i.A =
has order 2×2
ii.B =
has order 2×3
iii. C = (P Q) has order 1×2
iv. D =
has order 3×1
SPECIAL MATRICES
Is the matrix having all elements zero ( zero matrix)
Z =
IDENTITY MATRIX: Is the square matrix whose elements is the leading diagonal are everywhere 1 and 0 elsewhere.
I =
.=
Identity matrix
ADDITION OF MATRICES
Matrix addition is performed by adding corresponding elements.
for example
Example
1. Given matrices A = (1 2 3) and B = ( 4 5 6)
Find
i. A + B
ii. B + A
Solution
i. A + B = ( 1 2 3) + ( 4 5 6)
= ( 1+4 2+5 3+6)
= ( 5 7 9)
ii. B + A = ( 4 5 6 ) + ( 1 2 3 )
= ( 4+1 5+2 6+3)
= (5 7 9)
2. If A = and B =
Then, find A + B
A + B =+
=
1. Given matrices A = (1 2 3) and B = ( 4 5 6)
Find
i. A + B
ii. B + A
Solution
i. A + B = ( 1 2 3) + ( 4 5 6)
= ( 1+4 2+5 3+6)
= ( 5 7 9)
ii. B + A = ( 4 5 6 ) + ( 1 2 3 )
= ( 4+1 5+2 6+3)
= (5 7 9)
2. If A =
and B = 
Then, find A + B
A + B =
+ 
=
ADDITIVE IDENTITY MATRIX
Consider any 2 x 2 matrix N = , where a, b, c, and dare any real number. If N + Y = Y + N = N, then N
is the addictive identity matrix.
N = , where a, b, c, and dare any real number. If N + Y = Y + N = N, then N is the addictive identity matrix.
The 2 x 2 addictive identity matrix is
Let matrix A =
And Z =
Then Z is an additive identity matrix.
i.e. A+Z = A and Z +A = A
And Z =
Then Z is an additive identity matrix.
i.e. A+Z = A and Z +A = A
ADDITIVE INVERSE MATRIX
Consider any two matrices of the same order P and Q.
If P + Q = Q + P = R, then Q is called the additive inverse of P or P is called the additive inverse of Q.
i.e. Q = -P or P = -Q
Suppose,
Suppose,
If P + Q = Z then,
= -P
Therefore the additive inverse of P is -P
= -P
Therefore the additive inverse of P is -P
Example
1. Find the additive inverse of
(a) B = (b) C=
1. Find the additive inverse of
(a) B =
(b) C= 
Solution
SUBTRACTION OF MATRICES
The process of subtracting a real number “f” from another real number g is the same as adding g to the additive inverse of f.
Thus f-g = f + (-g).
NOTE
When matrix P is subtracted from another matrix Q the result is the same as adding P to the additive inverse of Q.
i.e P- Q = P + (-Q).
SCALAR MULTIPLICATION OF MATRICES
Example If B =
i. Find 2B
Solution
2
= 
ii. Find
B Solution
= 
Questions
1. Given;
A =
B = 
C = 
FIND.
(a) 3A + 2B
Solution
3A = 3
3A =
2B = 2
2B =
3A+ 2B =
=
(b) 5 ( A + B)
A+ B =
= 
5(A+B) = 5
=
2. Using the matrices;
A =
, B = 
and C = 
a) Find A (BC)
BC =
BC =
A (BC) =
=
=
b) (AB ) C
AB =
AB =
C =
(AB)C =
(AB)C =
DETERMINANT OF A MATRIX
NOTE
Determinants exist for square materials only.
Calculate the determinant of a matrix and tell whether the matrix is singular or non singular.
1. A =
Solution
= (-1 x4) – (2×3)= -4 -6
= -10
A is non singular matrix.
Inverse of matrices
The inverse of a matrix say P is another matrix denoted by P-1
NOTE
Can be found by interchanging the elements of the leading diagonal so that d takes place of a and a takes place of d. Change the sign of the elements in the main diagonal so that b and c becomes –b and –c respectively.
Divide each element by the determinant of A
2. Inverse exist for non singular matrix.
3. Singular matrix has no inverse because they have zero determinant.
Example
Determine the inverse of the gives matrix and indicate if it is singular or non singular.
1. A =
Solution
Determinant; (A) = (4×4) – (-4×4)
= 16+16
= 32
A-1 =
A-1 =
A is non singular matrix.
2. B =
Solution
Determinant ( B ) = (-1×1) – (-1x-1)
= -1-1
= -2
B -1 = -1/2
B-1 =
MATRICES ON SOLVING SIMULTANEOUS EQUATIONS
Questions
1.5X + 6Y = 1
7X + 8Y = 15
=
Let
be A
= (5×8) – (7×6)
= 40 – 42
= -2
A-1=
A-1 =
= =
= =
=
x = 1
y = 1
2.Solve the following simultaneous equation by matrix.
4X + 2Y= 40
X + 3Y = 35
=
Let be B
= (4×3 ) – ( 2×1)
= 12 2
= 10
B-1=
=
=
Example
Determine the inverse of the gives matrix and indicate if it is singular or non singular.
1. A =
Solution
Determinant; (A) = (4×4) – (-4×4)
= 16+16
= 32
A-1 =
A-1 =
A is non singular matrix.
2. B =
Solution
Determinant ( B ) = (-1×1) – (-1x-1)
= -1-1
= -2
B -1 = -1/2
B-1 =
MATRICES ON SOLVING SIMULTANEOUS EQUATIONS
Questions
1.5X + 6Y = 1
7X + 8Y = 15
= 
Let
be A 
= (5×8) – (7×6)= 40 – 42
= -2
A-1=
A-1 =
=
= =
= 
= 
x = 1
y = 1
2.Solve the following simultaneous equation by matrix.
4X + 2Y= 40
X + 3Y = 35
= 
Let
be B 
= (4×3 ) – ( 2×1)= 12
2= 10
B-1=
=
=