Tutorial 6 on Matrices PREPARATION
Questions 1 to 5 are base on the following information:
Define the following matrices:
2 5 0
4 15 5 𝑞 2 3 4
𝑇= ,𝐷 = ,𝑊 = 4 1 −3 and 𝑍 =
2 10 2 6 1 −1 0
−1 0 1
1. 𝑊 + 3𝐼 =
2. 𝑇𝑍 =
3. Calculate the value of 𝑞 that will make 𝐷 singular.
4. Calculate the determinant of 𝑊.
5. Investigate the following properties of determinants for matrix 𝑇:
5.1 Show that |𝑇| = |𝑇 |
5.2 Interchange row 1 and 2 of 𝑇, and show that the sign of the determinant changes but
not the numeric value.
5.3 Multiply the second column of 𝑇, with 4 and investigate whether the value of the
determinant changes 4 – fold.
5.4 Subtract each element of row 2 from the corresponding element of row 1, show that
the value of the determinant remains the same.
5.5 Substitute column 2 with 12 and 6, so that column 2 is a multiple of column 1. Show
that the determinant of the new matrix is zero.
Questions 6 to 10 are based on the following information:
A coffee shop supplies cocktail muffins, cocktail cupcakes and sausage rolls (per dozen) for
the year end functions of three different departments. The order (per dozen) for the functions
is as follows:
Refreshments (Quantities per dozen)
Department Cocktail Cocktail Total income
Sausage rolls
muffins cupcakes
Department A 4 6 10 580
Department B 0 9 20 910
Department C 10 4 0 320
Price (Rand per dozen) 𝒑𝟏 𝒑𝟐 𝒑𝟑
The prices (in Rand per dozen) of the refreshments are given by: 𝑷 = (𝒑𝟏 𝒑𝟐 𝒑𝟑 )
𝟒 𝟔
𝟏𝟎 𝟓𝟖𝟎
Let: 𝑨= 𝟎 𝟐𝟎 be the quantity matrix and 𝑩 = 𝟗𝟏𝟎 the vector that denotes the
𝟗
𝟏𝟎 𝟒
𝟎 𝟑𝟐𝟎
𝟏 𝟒 −𝟐 −𝟏. 𝟓
total income, 𝑬 = 𝟏 , |𝑨| = −𝟐𝟎 𝐚𝐧𝐝 𝑨 𝟏 = −𝟏𝟎 𝟓 𝟒
𝟏 𝟒. 𝟓 −𝟐. 𝟐 −𝟏. 𝟖
6. The total number of refreshments ordered per department in matrix notation is:
7. If the quatities ordered for Department A doubles and the quantities for Department B is
the same as for Department C, the determinant of the altered matrix 𝑨 is:
8. Write down the set of linear equations that can be used to calculate the prices (in Rand
per dozen) of the refreshments.
9. Calculate the price (in Rand per dozen) of cocktail muffins using Cramer’s rule.
10. Calculate the price (in Rand per dozen) of cocktail cupcakes using the inverse matrix
approach.
Questions 1 to 5 are base on the following information:
Define the following matrices:
2 5 0
4 15 5 𝑞 2 3 4
𝑇= ,𝐷 = ,𝑊 = 4 1 −3 and 𝑍 =
2 10 2 6 1 −1 0
−1 0 1
1. 𝑊 + 3𝐼 =
2. 𝑇𝑍 =
3. Calculate the value of 𝑞 that will make 𝐷 singular.
4. Calculate the determinant of 𝑊.
5. Investigate the following properties of determinants for matrix 𝑇:
5.1 Show that |𝑇| = |𝑇 |
5.2 Interchange row 1 and 2 of 𝑇, and show that the sign of the determinant changes but
not the numeric value.
5.3 Multiply the second column of 𝑇, with 4 and investigate whether the value of the
determinant changes 4 – fold.
5.4 Subtract each element of row 2 from the corresponding element of row 1, show that
the value of the determinant remains the same.
5.5 Substitute column 2 with 12 and 6, so that column 2 is a multiple of column 1. Show
that the determinant of the new matrix is zero.
Questions 6 to 10 are based on the following information:
A coffee shop supplies cocktail muffins, cocktail cupcakes and sausage rolls (per dozen) for
the year end functions of three different departments. The order (per dozen) for the functions
is as follows:
Refreshments (Quantities per dozen)
Department Cocktail Cocktail Total income
Sausage rolls
muffins cupcakes
Department A 4 6 10 580
Department B 0 9 20 910
Department C 10 4 0 320
Price (Rand per dozen) 𝒑𝟏 𝒑𝟐 𝒑𝟑
The prices (in Rand per dozen) of the refreshments are given by: 𝑷 = (𝒑𝟏 𝒑𝟐 𝒑𝟑 )
𝟒 𝟔
𝟏𝟎 𝟓𝟖𝟎
Let: 𝑨= 𝟎 𝟐𝟎 be the quantity matrix and 𝑩 = 𝟗𝟏𝟎 the vector that denotes the
𝟗
𝟏𝟎 𝟒
𝟎 𝟑𝟐𝟎
𝟏 𝟒 −𝟐 −𝟏. 𝟓
total income, 𝑬 = 𝟏 , |𝑨| = −𝟐𝟎 𝐚𝐧𝐝 𝑨 𝟏 = −𝟏𝟎 𝟓 𝟒
𝟏 𝟒. 𝟓 −𝟐. 𝟐 −𝟏. 𝟖
6. The total number of refreshments ordered per department in matrix notation is:
7. If the quatities ordered for Department A doubles and the quantities for Department B is
the same as for Department C, the determinant of the altered matrix 𝑨 is:
8. Write down the set of linear equations that can be used to calculate the prices (in Rand
per dozen) of the refreshments.
9. Calculate the price (in Rand per dozen) of cocktail muffins using Cramer’s rule.
10. Calculate the price (in Rand per dozen) of cocktail cupcakes using the inverse matrix
approach.
