MATRIX

A. Definition Of a Matrix

A matrix is an ordered set of numbers listed in a rectangular array, arrangged

in rows and columns. Every number in a matrix is called the element or the entry of
the matrix.

In daily life, information is often presented in the form of a table or list. For
example, the table of test result stated as in the following.

Name Math Physics Chemistry Biology

Amir 70 65 85 80
Budi 65 70 65 95
Candra 75 75 80 9
(Table 1)

If the data from table 1 above is written only as numbers, we will get an
ordered list of numbers endclosed in brackets.

70 65 85 80

65 70 65 95

75 75 80 90

The ordered set of numbers has a specific arrangement, i.e., arranged in a

square or rectangular array given in rows and columns. The ordered set of numbers
listed in a rectangular array is called a matrix

B. The Notation and the Orderof a Matrix

A matrix is usually notated by capital letters, while it’s element is usually

represented in lower case letters. For example,

70 65 85 80 2 3
a c
A(3×4) = 65 70 65 95 B(4×2)= -1 2 and C(2×2)= b d

75 75 80 90 0 1
 14 -3

The size ofamatrix is often called the order of the matrix. The order of a matrix is
determined by the number of rows and the number of columns in the matrix.

If matrix A consists of m rows and n columns, then order of matrix A is m × n

and written as: Am×n . The number of elements in matrix Ais (m × n). Therefore,
matrix A of order m × n can be presented as the following.

a11 a12 a13 … … a1n

a21 a22 a23 … … a2n

Am×n = … … … … … … numbers of row =

m

… … … … … …

am1 am2 am3 … … amn

number of columns = n

wheream×n is the matrix element which lies in the m-th row and n-th column.

C. Types of Matrix

 Rows of Matrix
A row matrix is a matrix which consists of one row. Generally, a row matrix
has order 1 × n.
MatricesP1 × 2 = (3 2) and Q1×3 = (2 -1 0), are example of row matrices.

 Column matrix
A column matrix is a matrix which consists of one column. Generally, a
column matrix has order m × 1.
3 3
Matrix X3 × 1 = -1 and Y4× 1 = 0 ,are example of column matrices.
 2 -2
5

 Squere matrix
A squere matrix is a matrix with the same number of rows and coloums.
Generally a squere matrix has ordere n × n.

−1 1 3
0 −1
Matrix R2×2 = and S3×3 = −2 4 7 , are example of squere matrices
2 4
3 5 1

 Transpose of a matrix
The transpose of matrix A, written as At or A` is a matrix obtained by turning
all the rows of matrix A into columns of A`. So, if given

a b a c

A= c d , then At = b d

D. Aljebraic Operations on Matrices

1. Matrix Addition

If A and B are two matrices of the same order, then the sum of matrices A and
B, written as A + B is a matrix obtainedby adding every corresponding element of A
and B.

Example :

Find the sum of the following matrices.

j k o p

A= l m B= q t

Solution :
 j k o p j+o k+p
A+B= + =
l m q t i+q m+t

Properties of matrix addition :

Let A, B and C be matrices of ordere m× n, then :

A + B = B + A (the commutative property)
this property allow us to interchange the order of operaition.
(A + B) + C = A + ( B + C) (the assosiative property)
This property allow us to write A + B + C with no other meaning.

2. Matrix Subtraction
If A and B are two matrices of the same order, then the subtraction of matrices
A and B can be exspressed as follows :
A - B = A+ ( ̶ B)
In this case, ̶ B is the opposite of matrix B.

Example :

a b j k
Given A = andP = .find matrix A ̶ P.
c d l m

Solution :
𝑎 𝑏 𝑗 𝑘 𝑎 𝑏 −𝑗 −𝑘
A ̶ P = A + (̶ P)↔ ̶ = +
𝑐 𝑑 𝑙 𝑚 𝑐 𝑑 −𝑙 −𝑚

𝑎−𝑗 𝑏−𝑘
=
𝑐−𝑙 𝑑−𝑚

3. Scalar Multiplication of a Matrix

 Let k be real number and A be a matrix, then kA is the matrix obtained
by multiplying every element of A by the scalar k. therefore :

𝑎 𝑏 𝑎 𝑏 𝑘×𝑎 𝑘×𝑏
If given A = , then kA= k =
𝑐 𝑑 𝑐 𝑑 𝑘×𝑐 𝑘×𝑑

Properties of scalar multiplication of a matrix

Let p, q,and r be real numbers, and A and B be matrices of order m×n, then :
1. (q + r)A = qA + rA
2. r(A + B)= rA + rB
3. p(qA) = (pq)A

4. Multiplication Matrix and Matrix

Matrix multiplication is process or matrix operation that is multiplying every

elements on the matrix row on the left by matrix column on the right, and then the
result are added.
Example :
𝑎 𝑏 𝑝 𝑞
If matrix A = and B = , then the product of A and B can be
𝑐 𝑑 𝑟 𝑠
determined by the question :

𝑎 𝑏 𝑝 𝑞 𝑎𝑝 + 𝑏𝑟 𝑐𝑝 + 𝑏𝑠
AB = =
𝑐 𝑑 𝑟 𝑠 𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠

Constraints subject to matrix multiplication :

In matrix multiplication, there are two things consider, which are :
1. The existential of the product of multiplication.
2. If the product exists, then how to find the order of the product of
multiplication.
The existential of a product of a matrix multiplication and the order of the product
matrix can be ilustrated by using a pair of domino cards :

Equal

2×3 3×1 a. The product of the

matrix
multiplication exists.
b. the order of the product
matrix is matrix is 2×1

2×1

unequal

a. The product of the

matrix multiplication
does not exist
(undefined)

In the figure, the domino card on left state the order of the left matrix and the
domino card on the right state the order of the right matrix. Based on the ilustration
above, we can conclude that the product of matrix multiplication A of order (m×k)
and B of order (k×n) is a matrix C of order (m×n).

A(m×k) × B(k×n) = C(m×n)

Example :

2 −1 4
Given matrices A= and B= . Find the product of matrix multiplication
1 3 5

Solution :

2 −1 4
A(2×2 ) = and B(2×1) =
1 3 5
 2 −14 (2 × 4) + (−1 × 5)
A(2×2 ) × B(2×1)= =
1 35 (1 × 4) + (3 × 5)

8 + (−5)
the order of the product matrix is (2×1) =
4 + 15

3
C(2×1) =
19

Law of a matrix multiplication :

1) The matrix multiplication is associative.

(AB)C = A(BC)
2) The matrix multiplication is distributive
Left distrbutive : A (B + C) = AB + AC
Right distrbutive : (A + B) C = AC + BC
3) The matrix multiplication is not commutative
AB ≠ BA

E. TheDeterminant ofMatrix of Order 2×2

Let A be a squere matrx of order 2×2, generally written as :

secondary diagonal
𝑎 𝑏
A=
𝑐 𝑑
main diagonal

The product of all elements on te main diagonal minus the product of all elements
on the secondary diagonal, i.e (𝑎𝑑 − 𝑐𝑑) is called the determinant of matrix A and
often denoted by det A or |𝐴|.
For example :
 2 1 2 1
P= , then det = (2 × 3) − (4 × 1) = 2
4 3 4 3

|𝑃| = 2 or detP = 2