Matrices and determinants

2000017610

Level:

For which positive real number

a

is the determinant of the matrix equal to

36

(\begin{matrix} 2 a & 0 & a \\ 0 & 1 & - 1 \\ a & 2 a & 6 \end{matrix})

a = 2

a = 4

a = 6

a = 0

2000018401

Level:

Assuming that

| \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ x & y & z \end{matrix} | = 5

compute the following determinant:

| \begin{matrix} a & b & c \\ x & y & z \\ 3 & 3 & 3 \end{matrix} | .

15

- 15

27

other

2000018402

Level:

Which two of the matrices,

A

B

, and

C

have the same determinant?

A = (\begin{matrix} 1 & 3 & 5 \\ 5 & 3 & 2 \\ 1 & 5 & 3 \end{matrix}), B = (\begin{matrix} 1 & 2 & 5 \\ 5 & 3 & 3 \\ 1 & 5 & 3 \end{matrix}), C = (\begin{matrix} 1 & 5 & 1 \\ 3 & 3 & 5 \\ 5 & 2 & 3 \end{matrix})

A

and

C

A

and

B

B

and

C

none

2000018403

Level:

Assuming that

| \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ x & y & z \end{matrix} | = 5

compute the following determinant.

| \begin{matrix} 5 & 5 & 5 \\ a + 2 & b + 2 & c + 2 \\ \frac{x}{3} & \frac{y}{3} & \frac{z}{3} \end{matrix} |

\frac{25}{3}

\frac{5}{3}

7

\frac{7}{3}

2000018404

Level:

Assuming that

| \begin{matrix} 4 & 0 & 2 \\ a & b & c \\ 2 & 4 & 6 \end{matrix} | = 36

compute the following determinant.

| \begin{matrix} 1 & 2 & 3 \\ a & b & c \\ 4 & 0 & 2 \end{matrix} |

- 18

18

72

- 72

2000018405

Level:

Assuming that

| \begin{matrix} 1 & a & 5 \\ 1 & b & 6 \\ 1 & c & 7 \end{matrix} | = - 4

compute the following determinant:

| \begin{matrix} a & - 2 & 6 \\ b & - 2 & 7 \\ c & - 2 & 8 \end{matrix} |

- 8

8

- 7

9

2000018406

Level:

Which matrix from

A

B

C

and

D

does not have a determinant equal to zero?

A = (\begin{matrix} 1 & 2 & 5 \\ 1 & 3 & 6 \\ 1 & 4 & 7 \end{matrix}),

B = (\begin{matrix} 1 & 2 & 3 \\ 0 & 1 & - 1 \\ 2 & 4 & 6 \end{matrix}),

C = (\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 15 & 16 & 17 \end{matrix}),

D = (\begin{matrix} 1 & 2 & 5 \\ 1 & - 4 & - 6 \\ 1 & - 4 & 7 \end{matrix})

D

A

B

C

2000018407

Level:

Compute the determinant of the following matrix:

A = (\begin{matrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ - 1 & 4 & 11 \end{matrix})

33

0

34

15

Matrices and determinants

2000017610

2000018401

2000018402

2000018403

2000018404

2000018405

2000018406

2000018407

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