Matrices and determinants

2000019302

Level:

Three ice cream stands of ICE company reported their July sales of four ice cream flavors in number of portions sold. All data can be seen in the table below:

\begin{array}{ccccc} vanilla & chocolate & nut & strawberry \\ Stand 1 & 720 & 800 & 1 200 & 360 \\ Stand 2 & 550 & 434 & 900 & 300 \\ Stand 3 & 610 & 300 & 200 & 750 \end{array}

The profits from the sales of each specific flavor are expressed by the matrix

P = (\begin{matrix} 1 \\ 1 \\ 3 \\ 2 \end{matrix})

. Let the July sales be rewritten to the matrix

J

. By what matrix the profits of individual ice cream stands in July are described?

J \cdot P

P \cdot J

J + P

Profits cannot be determined by any of the matrix operations.

2000019303

Level:

Three ice cream stands of ICE company reported their July sales of four ice cream flavors in number of portions sold. All data can be seen in the table below:

\begin{array}{ccccc} vanilla & chocolate & nut & strawberry \\ Stand 1 & 720 & 800 & 1 200 & 360 \\ Stand 2 & 550 & 434 & 900 & 300 \\ Stand 3 & 610 & 300 & 200 & 750 \end{array}

The profits from the sales of each specific flavor are expressed by the matrix

P = (\begin{matrix} 1 \\ 1 \\ 3 \\ 2 \end{matrix})

. Estimate what was the total profit of ICE company in July from all three stands.

more than

12 000 CZK

from

9 000 CZK

12 000 CZK

from

6 000 CZK

9 000 CZK

less than

6 000 CZK

Adam’s mom and Peter’s mom compete to see who can buy groceries cheaper. For summer weekend, they both want to buy butter, sugar, flour, and vanilla sugar and there are two stores available Aha and Praha.

The following tables show the quantity of items they both plan to buy and their prices in stores Aha and Praha.

\begin{array}{ccccc} \begin{matrix} Butter \\ (kg) \end{matrix} & \begin{matrix} Sugar \\ (kg) \end{matrix} & \begin{matrix} Flour \\ (kg) \end{matrix} & \begin{matrix} Vanila sugar \\ (number of pieces) \end{matrix} \\ Adam's mom & 0.5 & 2 & 1 & 8 \\ Peter's mom & 0.25 & 2 & 2 & 5 \end{array}

\begin{array}{ccc} Aha & Praha \\ Butter (per 1 kg) & 119.6 CZK & 159.6 CZK \\ Sugar (per 1 kg) & 12.5 CZK & 9.9 CZK \\ Flour (per 1 kg) & 10.9 CZK & 9.5 CZK \\ Vanilla sugar (per piece) & 5.9 CZK & 5.1 CZK \end{array}

What follows from the next product of the two matrices?

(\begin{matrix} 0.5 & 2 & 1 & 8 \\ 0.25 & 2 & 2 & 5 \end{matrix}) \cdot (\begin{matrix} 119.6 & 159.6 \\ 12.5 & 9.9 \\ 10.9 & 9.5 \\ 5.9 & 5.1 \end{matrix}) = (\begin{matrix} 142.9 & 149.9 \\ 106.2 & 104.2 \end{matrix})

For Adam’s mom, it is more convenient to shop at Aha, while for Peter’s mom, it is more convenient to shop at Praha.

For Adam’s mom, it is more convenient to shop at Praha, while for Peter’s mom, it is more convenient to shop at Aha.

Shopping at Aha is more convenient for both moms.

Shopping at Praha is more convenient for both moms.

2000019305

Level:

Mr. Wise buys fuel at three various gas stations – MOL, Shell and EuroOil. He always buys

25

liters of diesel, cafe latte and

2

nougat croissants.

Prices in CZK:

\begin{array}{cccc} Diesel (1 liter) & 1 Cafe Latte & 1 Croissant \\ MOL & 31.20 & 57 & 16.90 \\ Shell & 27.20 & 52 & 20 \\ EuroOil & 29.60 & 49 & 18.20 \end{array}

Consider the next product of the two matrices:

(\begin{matrix} 31.20 & 57 & 16.90 \\ 27.20 & 52 & 20 \\ 29.60 & 49 & 18.20 \end{matrix}) \cdot (\begin{matrix} 25 \\ 1 \\ 2 \end{matrix})

Choose the statement which is NOT true:

The most optimal prices are at the MOL gas station.

The most optimal prices are at the Shell gas station.

The least optimal prices are at the MOL gas station.

2010006701

Level:

Identify a true statement related to the following matrix

A

A = (\begin{matrix} 2 & 4 & - 3 & 7 \\ 9 & - 5 & - 1 & 8 \\ 11 & 0 & 8 & 12 \\ - 7 & - 8 & 1 & 13 \\ 9 & 10 & - 6 & 2 \end{matrix})

A

is a

5 \times 4

matrix and

a_{(2, 3)} = - 1

A

is a

5 \times 4

matrix and

a_{(2, 3)} = 0

A

is a

4 \times 5

matrix and

a_{(2, 3)} = 0

A

is a

4 \times 5

matrix and

a_{(2, 3)} = - 1

9000019903

Level:

Identify a true statement related to the following matrix

A

A = (\begin{matrix} - 2 & 3 & 10 & 5 & - 5 \\ 6 & 11 & - 7 & 2 & - 3 \\ - 7 & 15 & - 6 & 2 & 4 \\ - 8 & 1 & 13 & - 5 & 0 \end{matrix})

A

is a

4 \times 5

matrix and

a_{(3, 2)} = 15

A

is a

4 \times 5

matrix and

a_{(2, 3)} = 15

A

is a

5 \times 4

matrix and

a_{(3, 2)} = - 7

A

is a

5 \times 4

matrix and

a_{(3, 2)} = 15

2000017101

Level:

Determine all real numbers

b

such that there exists the inverse matrix to the matrix:

(\begin{matrix} 4 & 3 & b \\ 2 & 1 & 2 \\ b & b & - 1 \end{matrix})

all real numbers

all non-negative real numbers

all positive real numbers

Such a number does not exists.

2000017102

Level:

Compute the inverse matrix to the matrix:

(\begin{matrix} 4 & 3 & 0 \\ 2 & 1 & 2 \\ 0 & 0 & - 1 \end{matrix})

(\begin{matrix} - \frac{1}{2} & \frac{3}{2} & 3 \\ 1 & - 2 & - 4 \\ 0 & 0 & - 1 \end{matrix})

(\begin{matrix} \frac{1}{2} & \frac{3}{2} & 3 \\ 1 & - 2 & - 4 \\ 0 & 0 & 1 \end{matrix})

(\begin{matrix} - \frac{1}{2} & \frac{3}{2} & - 3 \\ - 1 & - 2 & - 4 \\ 0 & 0 & - 1 \end{matrix})

(\begin{matrix} - \frac{1}{2} & \frac{3}{2} & 3 \\ 1 & 2 & - 4 \\ 0 & 0 & - 1 \end{matrix})

2000017103

Level:

Compute the inverse matrix to the matrix:

(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ - 1 & 0 & 1 \end{matrix})

(\begin{matrix} 1 & 0 & 0 \\ - 1 & 1 & - 1 \\ 1 & 0 & 1 \end{matrix})

(\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{matrix})

(\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ - 1 & 0 & - 1 \end{matrix})

(\begin{matrix} - 1 & 0 & 0 \\ - 1 & 1 & - 1 \\ 1 & 0 & - 1 \end{matrix})

2000017104

Level:

Find the number

a

so that the matrices given below are inverse to each other.

(\begin{matrix} 1 & 2 \\ 2 & 2 \end{matrix}), (\begin{matrix} - 1 & 1 \\ 1 & a \end{matrix})

a = - \frac{1}{2}

a = \frac{1}{2}

a = 0

a

cannot be found.

Matrices and determinants

2000019302

2000019303

2000019304

2000019305

2010006701

9000019903

2000017101

2000017102

2000017103

2000017104

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