Matrices and determinants

2000017201

Level:

Which of the given matrices is of order

3

and has an entry of

2

at the position

(3, 2)

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

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

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

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

2000017202

Level:

Which of the given matrices

K

L

M

, and

N

have the same main diagonal?

K = (\begin{matrix} 1 & 7 & 8 \\ 4 & 2 & 9 \\ 5 & 6 & 3 \end{matrix}), L = (\begin{matrix} 3 & 9 & 8 \\ 4 & 2 & 7 \\ 5 & 6 & 1 \end{matrix}),

M = (\begin{matrix} 1 & 7 & 9 \\ 5 & 2 & 8 \\ 4 & 6 & 3 \end{matrix}), N = (\begin{matrix} 2 & 7 & 8 \\ 4 & 2 & 6 \\ 5 & 7 & 5 \end{matrix})

K

and

M

K

L

and

M

K

L

and

N

The main diagonals of any two matrices differ.

2000017203

Level:

Which of the given matrices have the same entry at the position

(1, 2)

K = (\begin{matrix} 1 & \sqrt{2} & 3 & \sqrt{5} \\ \sqrt{3} & 2 & 1 & 5 \\ 4 & 1 & 1 & 0 \end{matrix}), L = (\begin{matrix} 1 & \sqrt{2} & 3 \\ \sqrt{3} & 2 & 1 \\ 4 & 1 & 1 \\ \sqrt{5} & 5 & 0 \end{matrix}),

M = (\begin{matrix} \sqrt{3} & 2 & 1 \\ \sqrt{3} & 4 & 0 \end{matrix}), N = (\begin{matrix} 1 & \sqrt{3} & 4 \\ \sqrt{3} & 2 & 1 \\ 3 & 1 & 1 \\ \sqrt{5} & 5 & 0 \end{matrix})

K

and

L

K

L

and

N

K

L

M

, and

N

L

and

N

2000017204

Level:

Which of the given matrices is the matrix

(m_{i, j})

, where

i = 1, \dots, 3

and

j = 1, 2

(\begin{matrix} 8 & 7 \\ 6 & 5 \\ 4 & 3 \end{matrix})

(\begin{matrix} 8 & 7 & 6 \\ 5 & 4 & 3 \end{matrix})

(\begin{matrix} 8 & 7 & 6 \\ 5 & 4 & 3 \\ 2 & 1 & 0 \end{matrix})

(\begin{matrix} 8 & 7 \\ 6 & 5 \end{matrix})

2000017205

Level:

Which of the given matrices is the matrix

(m_{i, j})

, where

i = 1, \dots, 3

and

j = 1, \dots, 3

with

m_{i, j} = i + j + 1

(\begin{matrix} 3 & 4 & 5 \\ 4 & 5 & 6 \\ 5 & 6 & 7 \end{matrix})

(\begin{matrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{matrix})

(\begin{matrix} 3 & 4 & 5 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{matrix})

(\begin{matrix} 2 & 3 & 4 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{matrix})

2000017207

Level:

For what values of

a

and

b

is the matrix

(\begin{matrix} a + 5 & a + 3 & 0 \\ 0 & a - 2 & 0 \\ - a - b & a + b & a + 1 \end{matrix})

a diagonal matrix?

a = - 3

b = 3

a = 2

b = - 2

a = 3

b = - 3

a = - 3

b = - 3

2000017401

Level:

Find the product of the following matrices:

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

A B = (\begin{matrix} \frac{3}{4} & \frac{1}{4} \\ - \frac{3}{2} & 1 \end{matrix})

A B = (\begin{matrix} \frac{3}{4} & - \frac{3}{16} \\ 0 & \frac{3}{4} \end{matrix})

A B = (\begin{matrix} \frac{3}{4} & \frac{1}{2} \\ - \frac{3}{2} & 1 \end{matrix})

A B = (\begin{matrix} 2 & \frac{1}{2} \\ - 1 & 2 \end{matrix})

2000017402

Level:

Find the product:

(\begin{matrix} 1.2 & 1 & 0 \\ 0.3 & 0.1 & 1 \\ 2 & 0 & 4 \end{matrix}) \cdot (\begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{matrix})

(\begin{matrix} 1.2 & 2.2 & 1 \\ 1.3 & 1.4 & 1.1 \\ 6 & 6 & 4 \end{matrix})

(\begin{matrix} 1.2 & 1 & 0 \\ 0 & 0.1 & 1 \\ 2 & 0 & 4 \end{matrix})

(\begin{matrix} 3.2 & 3.2 & 1 \\ 1.4 & 1.4 & 1.4 \\ 6 & 7 & 4 \end{matrix})

(\begin{matrix} 1.2 & 2.2 & 1 \\ 1.1 & 1.4 & 0 \\ 2 & 4 & 6 \end{matrix})

2000017403

Level:

For which real numbers

a

b

c

d

is the following equality true?

2 \cdot (\begin{matrix} a & c \\ b & d \end{matrix}) - (\begin{matrix} 2 & 1 \\ 3 & 5 \end{matrix}) = (\begin{matrix} 4 & 7 \\ - 5 & - 5 \end{matrix})

a = 3

b = - 1

c = 4

d = 0

a = 1

b = 1

c = 4

d = 1

a = 3

b = - 4

c = 4

d = 0

a = 3

b = - 1

c = 4

d = 1

2000017404

Level:

Consider the matrices:

A = (\begin{matrix} 1 & - 1 & 2 \\ 3 & 4 & 1 \\ 0 & 1 & 0 \end{matrix}), B = (\begin{matrix} 2 & 3 & - 1 \\ - 2 & 1 & 1 \\ 1 & 4 & - 1 \end{matrix}) .

Find the transpose of the matrix

A + B

(\begin{matrix} 3 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & - 1 \end{matrix})

(\begin{matrix} 3 & 2 & 1 \\ 1 & 5 & 2 \\ 1 & 5 & - 1 \end{matrix})

(\begin{matrix} 3 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & 1 \end{matrix})

(\begin{matrix} 3 & 2 & 1 \\ 2 & 5 & 2 \\ 1 & 2 & - 1 \end{matrix})

Matrices and determinants

2000017201

2000017202

2000017203

2000017204

2000017205

2000017207

2000017401

2000017402

2000017403

2000017404

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