Evaluation of determinants and area of triangles using determinants - One Line Questions
1.
If det(A) = -3, what is det(-2A) for a 3x3 matrix A? —
24
2.
Evaluate the determinant of the matrix [[1, 1, 1], [a, b, c], [a^2, b^2, c^2]]. —
(a-b)(b-c)(c-a)
3.
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by 1/2 |Δ|, where Δ is the determinant of the matrix: —
[[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]]
4.
What is the value of det([[1, 2, 3], [4, 5, 6], [7, 8, 9]])? —
0
5.
If A is an orthogonal matrix, then det(A) = ? —
±1
6.
What is the determinant of the matrix [[1, sin θ, 1], [-sin θ, 1, sin θ], [1, sin θ, 1]]? —
0
7.
If det(A) = 1 and det(B) = 1, what is det(AB)? —
1
8.
What is the determinant of an identity matrix of any order? —
1
9.
If the three vertices of a triangle are (1, 2), (3, 4), and (5, 6), what is the value of the determinant used to find its area? —
0
10.
What is the determinant of a skew-symmetric matrix of odd order? —
0
11.
What is the determinant of the matrix [[cos θ, -sin θ], [sin θ, cos θ]]? —
1
12.
If the points (a, b), (c, d), and (e, f) are collinear, then the determinant of the matrix [[a, c, e], [b, d, f], [1, 1, 1]] is: —
0
13.
What is the area of the triangle formed by the lines x=0, y=0, and x+y=1? —
1/2
14.
If any two rows or columns of a determinant are identical, what is the value of the determinant? —
0
15.
If the vertices of a triangle are (0, 0), (a, b), and (c, d), its area is: —
1/2 |ad - bc|
16.
To calculate the area of a triangle with vertices (2, 3), (4, 1), and (6, 5) using determinants, the expression will involve: —
1/2 |det([[2, 3, 1], [4, 1, 1], [6, 5, 1]])|
17.
The area of a triangle with vertices A(x1, y1), B(x2, y2), C(x3, y3) can be expressed using determinants as: —
1/2 |det([[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]])|
18.
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be calculated using the determinant formula: —
1/2 |det([[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]])|
19.
Consider a triangle with vertices O(0,0), A(x1, y1), and B(x2, y2). Its area is given by: —
1/2 |x1y2 - x2y1|
20.
Let A be a 4x4 matrix with det(A) = 5. What is det(3A)? —
405
21.
If A is a 2x2 matrix with det(A) = 3, what is det(5A)? —
75
22.
If det(A) = 10 and det(B) = 5, what is det(A/B)? (Assuming B is invertible) —
2
23.
If A is a 3x3 matrix and det(A) = 10, what is det(A^3)? —
1000
24.
If A is a 3x3 matrix and det(A) = 2, what is det(2A)? —
8
25.
If A is a 3x3 matrix with det(A) = 5, what is det(A^-1)? —
1/5
26.
If det(A) = 5, what is det(A^T)? —
5
27.
What is the determinant of the matrix [[1, 2, 3], [0, 4, 5], [0, 0, 6]]? —
24
28.
If det(A) = 7, what is det(A^-1)? —
1/7
29.
If det(A) = 0, then the system of linear equations Ax = b has: —
No solution or infinitely many solutions
30.
What is the determinant of the matrix [[a+b, a, b], [b, b+a, a], [a, b, a+b]]? —
2(a+b)^3
31.
For a 3x3 matrix [[a, b, c], [d, e, f], [g, h, i]], which of the following is the correct expansion of its determinant? —
a(ei - fh) - b(di - fg) + c(dh - eg)
32.
What is the determinant of the matrix [[a, 0, 0], [0, b, 0], [0, 0, c]]? —
abc
33.
What is the value of the determinant of a 2x2 matrix [[a, b], [c, d]]? —
ad - bc
34.
If det(A) = 0, then matrix A is: —
A singular matrix
35.
The area of the triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by the absolute value of: —
1/2 * det([[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]])
36.
For two square matrices A and B of the same order, which property holds true? —
det(AB) = det(A) * det(B)
37.
If matrix B is obtained from matrix A by interchanging two adjacent rows, what is the relationship between det(B) and det(A)? —
det(B) = -det(A)
38.
If a scalar k is multiplied by a matrix A of order nxn, what is the relationship between det(kA) and det(A)? —
det(kA) = k^n * det(A)
39.
Which property of determinants states that if a determinant has a row or column consisting entirely of zeros, its value is zero? —
Row/column of zeros
40.
If a row (or column) of a determinant is multiplied by a scalar k, how does the value of the determinant change? —
It is multiplied by k
41.
If a matrix A is obtained by multiplying each element of a matrix B (of order nxn) by a scalar k, then det(A) = ? —
k^n * det(B)
42.
If the area of a triangle with vertices (1, 2), (3, k), and (5, 7) is 10 square units, find the value(s) of k. —
k = -3 or k = 11
43.
If A is a 3x3 matrix and det(A) = k, what is det(adj(A))? —
k^(n-1) where n=3
44.
If det(A) = 0, then the rows (or columns) of A are: —
Linearly dependent
45.
If A is a square matrix such that A^2 = A, then det(A) can be: —
0 or 1
46.
What is the determinant of a diagonal matrix? —
The product of its diagonal elements
47.
What does a determinant value of 0 for a triangle's vertices indicate? —
The vertices are collinear
48.
The determinant of a matrix is unchanged if: —
A multiple of one row is added to another row
49.
The value of the determinant |x y z|, |x^2 y^2 z^2|, |x^3 y^3 z^3| is: —
xyz(x-y)(y-z)(z-x)