(1)

Algebra Unit (1): Matrices

Definition: the matrix is a method for organizing data or tabulating a group of elements as horizontal rows and vertical columns.

Remarks: • The elements of a matrix are inserted between ( ) • If the number of rows = m and number of columns = n so the

dimensions (order) of the matrix is m × n where m and n are +ve integers, which is read "m by n"

• We use capital letters to name the matrix such as A , B , C ….. • The elements of a matrix denoted by small letters as a , b , c …….. • If A is a matrix its dimensions is m × n we can put it in the form

A = ( amn ) where m = 1 , 2 , 3 , ….. , m and n = 1 , 2 , 3 , ….. , n

Ex (1): A factory producing TV sets some of them are color TV and the other is B/W with three different sizes 14 inches , 17 inches and 21 inches

14 inches 17 inches 21 inches

Color TV 20 12 6 B/W TV 15 18 10

We can represent these data simply as follow: 20 12 615 18 10

Ex (2): list the matrix A = ( amn ) where m = 1 , 2 , 3 and n = 1 , 2

Solution

A = 11 12

21 22

31 32

a aa aa a

it's a matrix of dimensions 3 × 2

Ex (3): list the matrix B = ( bmn ) where m = 1 and n = 1 , 2 , 3

Solution

B = ( )11 12 13b b b it's a matrix of dimensions 1 × 3

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(2) Some special matrices:

1. Row matrix : is a matrix with only one row → A = ( 7 4 2 )

2. Column matrix: is a matrix with only one column → B = 2

5 8

−

3. Square matrix: is a matrix with an equal number of rows and columns

→ 2 -115 3

4. Zero matrix (null matrix): is a matrix that has all its elements zero

→ 0 0 00 0 0

it can be represented as 2 × 3

------------------------------------------------------------------------------------------------ Matrix transpose: A matrix which is formed by turning all the rows of a given matrix into

columns and vice-versa. The transpose of matrix A is written AT Note: (AT )T = A Ex (4): Find the transpose for each matrix of the following

A = 3 1 25 2 7

−

, B = ( )1 - 4 5 , C = 4 3

9 2−

−

Solution

AT = 3 51 22 7

−

, BT = 1

5 4

−

, CT = 4 9

3 2−

−

------------------------------------------------------------------------------------------------ Equality of two matrices:

Two matrices is said to be equal if and only if they have the same dimensions and the corresponding elements are equal

Ex (5): find the values of x and y if: x 4 2 3 25 11 5 2y+7

+ − − = −

Solution x 4 3 x 1+ = → = − 2y 7 11 2y 18 y 9+ = − → = − → = −

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(3) Exercise (1)

1. A textile factory produces three types of textiles, and this factory has 2 branches A & B. if the numbers of clothes produced as follow: Branch A: produces 60 clothes of wool, 40 of silk and 20 of cotton Branch B: produces 45 clothes of wool, 32 of silk and 85 of cotton Put the following data in the form of a matrix "X" using two methods

2. In a contest among three students, we get the following results: The 1st student answered 15 questions in math, 20 questions in science. The 2nd student answered 24 questions in math, 15 questions in science. The 3rd student answered 30 questions in math, no questions in science.

a. Write these data in the form of a matrix with dimensions 3 × 2 b. If 5 marks were assigned for each question, write another matrix

with the same dimensions represents the marks which of the students get in the two subjects

3. If A = 3 49 02 6

−

, B = x 1 y9 02 6

−

. complete the following:

a. The element which lies in the 2nd row and the 1st column of matrix B is …………….

b. The element b12 = …………. And the element a21 = …………. c. If A = B so x = …… and y = ……

4. If A = 1 4

2 3−

− and B =

1

30 8

1 4 5

. Find AT and BT

5. If 2 7 1x 3 1

2y 1 4 6 4

− + − = − . Find the possible values for x and y

6. Write a zero (null) matrix with dimensions 3 × 3

7. Find x , y and z which makes x 3y y 0 1y z 5 4 5

− = +

8. Find x , y and z which makes

z x 1 4 2 1 1 y 5 1 z 1 5

2 9 1 2 9 1

− = − −

9. Prove that for all values of x and y the following equality is false: 2x y 3y 5 9 3 x y 3 2

+ = −

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