Which one of the following statements is true?
a) For all matrices A and B, if AB = I then BA = I
b) If A is a 3 \times 3 matrix, then det(2A) = 8 det(A).
c) For all matrices A and B, det(AB) = det(A) det(B).
d) If B is obtained by interchanging two rows of A, then det(B) = -1.
not (a) since in general, AB≠BA
not (b) since multiplying a row by n multiplies the determinant by n.
So, |nA| = n^2 |A|
not (d), since interchanging two rows makes |B| = -|A|
Better review the topic again.
The true statement among the options is:
c) For all matrices A and B, det(AB) = det(A) det(B).
This is known as the multiplicativity property of the determinant.
To determine which of the statements is true, let's break down each option and explain how to approach them:
a) For all matrices A and B, if AB = I then BA = I:
To determine if this statement is true, we can use a counterexample. Consider the matrices:
A = [[1, 0], [0, 0]]
B = [[0, 0], [0, 1]]
AB = [[0, 0], [0, 0]]
BA = [[0, 0], [1, 0]]
Since AB ≠ I, this statement is false.
b) If A is a 3x3 matrix, then det(2A) = 8 det(A):
To verify this statement, we can use the property that the determinant of a scalar multiple of a matrix is equal to the determinant of the original matrix multiplied by that scalar. So, we have:
det(2A) = 2^3 det(A)
= 8 det(A)
Since the left side of the equation is equal to the right side, this statement is true.
c) For all matrices A and B, det(AB) = det(A) det(B):
To determine if this statement is true, we can use a counterexample. Consider matrices:
A = [[1, 0], [0, 0]]
B = [[0, 0], [0, 1]]
det(AB) = det([[0, 0], [0, 0]]) = 0
det(A) det(B) = (1 * 0) = 0
Since det(AB) = det(A) det(B), this statement is true.
d) If B is obtained by interchanging two rows of A, then det(B) = -1:
To verify this statement, we can note that when two rows of a matrix are interchanged, the determinant changes its sign. So, if A is a matrix and B is obtained by interchanging any two rows of A, then det(B) = -det(A). Thus, the given statement is true.
In summary:
a) False
b) True
c) True
d) True