Matrices

Matrices

Introduction
Matrices are used much more in daily lives than you might have thought. In fact they are in front of you every day, at school, and even at home; for example, i
organising learners for a parade, altering shapes of objects in video games, studyin
trends of businesses, and also in construction of buildings.
Understanding this topic will enable you to represent data in matrix form and us matrices as tools to solve problems.

8.1 Defining a Matrix and Stating its Order
Defining a matrix
Activity 8.1 (a) (Work in groups)
Look around your classroom environment.
(a) Count and record the numbers of rows and columns in your classroom.
(b) What other items are arranged in the form of rows and columns in your society?
Numbers / items can be arranged in rows and columns to form either a rectangular or a square array.
Stating the order of a matrix
Activity 8.1 (b) (Work in groups)

8.2 Knowing when Matrices can be Added or Multiplied
It is important that you learn about when matrices can be added or multiplied. The orders of matrices in a sum or product expression, determine whether the sum or product is possible or not.
Addition / subtraction of matrices.

Activity 8.2(a) (Work in groups)
Mr Kamuntu owns a boutique. His sales for dresses and shirts for two weeks are as
given in the table below

(a) For the first week, write the daily sales as a matrix. Do the same for the second week. State the order of each matrix formed.
(b) Write matrix for the total sales and state its order.
(c) What conclusion can you draw about matrices in (a) and (b)?

Multiplication of matrices
Activity 8.2 (b) (Work in groups)
Kataza is a Senior Three learner at Hope Secondary School. During holidays, she sells fruits at the local market to raise money for her scholastic materials and other personal requirements. She majorly deals in apples, mangoes and pawpaws. She sells each fruit at UGX 1,000, UGX 500, and UGX 3,000, respectively. In a particular week, she recorded her daily sales as shown in the table on the next page.

(a) Write down the matrix for the selling prices of all the fruits.
(b) Write down the matrix for the numbers of fruits sold in the 4 days.
(c) Find the total sales for the 4 days.

8.3 Finding the Determinant of a 2×2 Matrix
Activity 8.3 (Work in groups)
Given matrix A =
(a) study and state the order of matrix A.
(b) find the product of the;
i) first and last entries of matrix A
ii) second and third entries of matrix A
(c) determine the difference between the value in (b)(ii) from that in (b)(i). Hence, state the determinant of A

• The diagonal of the first and last entries is known as the major / leading diagonal.
• The diagonal of the second and third entries is known as the minor diagonal.

8.4 Finding the Inverse of a 2×2 matrix and
understanding why it may not be possible to do so
For numbers a and b, if a x b = 1, then b is the multiplicative inverse of a. You note
that ax 1 = a and b xl = b.
The inverse of a 2 x 2 matrix
Activity 8.4(a) (Work in group)

Matrices
8.5 Applying the Knowledge of Matrices in Solving Problems from Real-life Situations Problems in real-life situations that can be modelled into systems of linear equations can be solved using the knowledge of matrices. Commerce is one of the fields where systems of linear equations can be generated.

Activity 8.5 (Work in groups)
Aguti was puzzled on the individual prices of a wristwatch and a money purse, having listened to a narration of Keba and Niwe. From the same shop, Keba and Niwe bought a money purse and 2 wristwatches at UGX 45,000, and 2 money purses and a wristwatch at UGX 30,000, respectively.
(a) Form the linear equations for the purchases of Keba and Niwe.
(b) Write the equations in matrix form.
(c) Determine the inverse of the coeffcient matrix formed in (b).

3) Shapira bought five books and three pens from Shoprite. She spent UGX 6,200 altogether. Sandra, the sister, bought six books and four pens. She spent UGX 7,600 altogether.
(a) Find the cost of each book and pen.
(b) Shadrack, a brother to both, bought eight books and four pens. How much did he spend?
4) Two women, Anna and Babirye, went to a wholesale shop in Kampala. Anna bought five bags of posho and three bags of rice while Babirye bought four bags of posho and two bags of rice.
Altogether, Anna and Babirye spent UGX 950,000 and UGX 700,000, respectively.
(a) Find the cost of each bag of posho and rice.
(b) How many bags of rice and posho can one buy with UGX 500,000?
5) A man bought a television set and a home theatre at UGX 750,000. Another man bought two television sets and three home theatres at UGX 1,800,000. How much did each of these items cost?