Determinants of order two and three - One Line Questions
1.
If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], what is det(A) * det(B)? —
-24
2.
If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], what is det(AB)? —
-2
3.
If A = [[1, 2], [3, 4]] and B = [[1, 0], [0, 1]], what is det(A)? —
-2
4.
If A = [[a, b], [c, d]], then det(A^T) is equal to: —
det(A)
5.
What is the determinant of the matrix [[1, 1, 1], [a, b, c], [a^2, b^2, c^2]] (Vandermonde determinant)? —
(b-a)(c-b)(a-c)
6.
What is the determinant of the identity matrix of order 2, I = [[1, 0], [0, 1]]? —
1
7.
What is the determinant of the identity matrix of order 3, I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]? —
1
8.
What is the determinant of the matrix [[5, 0], [0, 5]]? —
25
9.
What is the value of the determinant of the matrix [[1, 0, 0], [2, 1, 0], [3, 4, 1]]? —
1
10.
What is the determinant of the matrix [[sin(theta), cos(theta)], [-cos(theta), sin(theta)]]? —
sin^2(theta) + cos^2(theta)
11.
What is the determinant of the matrix [[1, 1, 1], [1, 2, 3], [1, 4, 9]]? —
2
12.
What is the value of the determinant [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]? —
cos^2(theta) + sin^2(theta)
13.
What is the determinant of the matrix [[1, 2, 3], [1, 2, 3], [4, 5, 6]]? —
0
14.
What is the determinant of the matrix [[1, 0, 5], [0, 1, 6], [0, 0, 1]]? —
1
15.
What is the determinant of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] after performing the operation R2 -> R2 + 2R1? —
0
16.
What is the determinant of the matrix [[1, 0, 0], [0, 1, 0], [0, 0, 1]]? —
1
17.
What is the determinant of the matrix [[1, 2], [2, 4]]? —
0
18.
What is the determinant of the matrix [[1, 0, 0], [2, 3, 0], [4, 5, 6]]? —
18
19.
If A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is the determinant of A? —
-19
20.
If any two rows (or columns) of a 3x3 determinant are identical, what is the value of the determinant? —
0
21.
What is the determinant of a matrix where one row is a scalar multiple of another row? —
0
22.
If A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], what is the determinant of A? —
0
23.
What is the determinant of the matrix [[0, 0, 0], [1, 2, 3], [4, 5, 6]]? —
0
24.
If A = [[1, 2, 3], [0, 0, 0], [4, 5, 6]], what is det(A)? —
0
25.
If det(A) = 3 for a 3x3 matrix A, what is det(A^T)? —
3
26.
What is the determinant of the matrix [[1, 2], [3, 4]]? —
-2
27.
If A is a 2x2 matrix with det(A) = 7, what is det(2A)? —
28
28.
If det(A) = 5 for a 2x2 matrix A, what is det(3A)? —
45
29.
If A = [[2, 3], [4, 5]], what is the determinant of A, denoted as |A|? —
-2
30.
If A is a 3x3 matrix and det(A) = 10, what is det(2A)? —
80
31.
If A = [[2, 1], [1, 2]], what is det(A^2)? —
9
32.
If A is a 2x2 matrix with det(A) = -3, what is det(-A)? —
9
33.
If det(A) = 5 and A is a 2x2 matrix, what is det(A^-1)? —
1/5
34.
If A = [[2, 0], [0, 3]], what is det(A^3)? —
216
35.
The determinant of a diagonal matrix of order 3, [[a, 0, 0], [0, b, 0], [0, 0, c]], is: —
abc
36.
For a 3x3 matrix A, if det(A) = 0, what does this imply about the matrix A? —
A is singular
37.
What is the value of the determinant of a 3x3 matrix [[a, b, c], [d, e, f], [g, h, i]]? —
a(ei - fh) - b(fg - di) + c(dh - eg)
38.
If A = [[a, b], [c, d]] and det(A) = 0, then which of the following is true? —
ad = bc
39.
If A = [[a, b], [c, d]], then det(A) = 0 implies: —
The rows (or columns) are linearly dependent
40.
What is the determinant of the matrix [[a, b], [c, d]] if a, b, c, d are in arithmetic progression? —
0
41.
What is the value of the determinant of a 2x2 matrix [[a, b], [c, d]]? —
ad - bc
42.
If A = [[a, b], [c, d]] and B = [[e, f], [g, h]], then det(AB) is equal to: —
det(A) * det(B)
43.
If the elements of a determinant are all integers, what can be said about the value of the determinant? —
It is always an integer
44.
What is the determinant of the matrix [[a, b, c], [ka, kb, kc], [d, e, f]]? —
0
45.
If A = [[a, b], [c, d]] is a 2x2 matrix, then det(kA) for a scalar k is equal to: —
k^2 * det(A)
46.
If a matrix A is multiplied by a scalar k, then det(kA) for a 3x3 matrix A is equal to: —
k^3 * det(A)
47.
For a 2x2 matrix A = [[p, q], [r, s]], if ps - qr = 0, what is the relationship between the rows or columns? —
Rows are proportional
48.
If A = [[a, b], [c, d]] is a 2x2 matrix and det(A) = 0, what can be said about the vectors [a, c] and [b, d]? —
They are linearly dependent
49.
What is the determinant of the matrix [[x, y], [y, x]]? —
x^2 - y^2
50.
What is the determinant of the matrix [[x, y, z], [x, y, z], [a, b, c]]? —
0