If A is a 3x3 matrix with determinant det(A) = 5, what is det(2A)?
A
10
B
40
C
20
D
8
Analysis & Theory
For a 3x3 matrix, det(kA) = k^3 * det(A). Here, 2^3 * 5 = 8*5 = 40.
If A and B are 3x3 matrices, which of the following is true?
A
det(A+B) = det(A) + det(B)
B
det(AB) = det(A) * det(B)
C
det(A-B) = det(A) - det(B)
D
det(AB) = det(A) + det(B)
Analysis & Theory
Determinant of a product of matrices equals the product of determinants.
Find the determinant of the matrix: [[1,2,3],[0,1,4],[5,6,0]]
A
1
B
24
C
1
D
-1
Analysis & Theory
Use the formula for 3x3 determinant: det = a(ei − fh) − b(di − fg) + c(dh − eg) = -1.
If A is a 3x3 identity matrix, det(A) = ?
A
0
B
1
C
3
D
-1
Analysis & Theory
Determinant of an identity matrix is always 1.
If two rows of a 3x3 matrix are equal, the determinant is:
A
1
B
0
C
Depends on matrix
D
-1
Analysis & Theory
If two rows are identical, the determinant of the matrix is zero.
If A is a 3x3 matrix and det(A) = -3, what is det(-A)?
A
3
B
9
C
-3
D
-9
Analysis & Theory
For 3x3 matrix, det(-A) = (-1)^3 * det(A) = -1 * -3 = 3.
Find the determinant of [[2,0,1],[1,3,2],[0,5,1]]
A
-20
B
10
C
5
D
15
Analysis & Theory
Using the 3x3 determinant formula, det = 10.
If A is a 3x3 matrix, det(A^T) = ?
A
det(A)
B
-det(A)
C
0
D
Cannot say
Analysis & Theory
Determinant of a matrix equals the determinant of its transpose.
If A is a 3x3 matrix and det(A) = 7, what is det(3A)?
A
63
B
189
C
21
D
7
Analysis & Theory
det(kA) = k^3 * det(A) = 3^3 * 7 = 27*7 = 189.
If det(A) = 5 and det(B) = -2 for 3x3 matrices A and B, find det(2AB)
A
-40
B
-80
C
20
D
40
Analysis & Theory
det(2AB) = 2^3 * det(A) * det(B) = 8 * 5 * -2 = -80.