Matrices & Determinants
Matrix multiplication
Grade Class 12
Question:
If A and B are two 3 x 3 matrices such that their product AB is a null matrix then is/are always true (where X<sup>T</sup> denotes transpose of X and I denotes unit matrix)-<br>(A) det. A ≠ 0 ⇒ B must be a null matrix.<br>(B) det. B ≠ 0 ⇒ A must be a null matrix.<br>(C) If none of A and B are null matrices then atleast one of the two matrices must be singular.<br>(D) If neither det. A nor det. B is zero then the given statement is not possible.
(A) det. A ≠ 0 ⇒ B must be a null matrix.
(B) det. B ≠ 0 ⇒ A must be a null matrix.
(C) If none of A and B are null matrices then atleast one of the two matrices must be singular.
(D) If neither det. A nor det. B is zero then the given statement is not possible.
Step-by-Step Solution
Key Concept: If AB = 0, then either A or B must be singular (det(A)=0 or det(B)=0). If det(A) != 0, then A is invertible, so A^-1(AB) = A^-1(0) => B = 0. Similarly for det(B) != 0.
If AB = 0, then det(AB) = det(A)det(B) = 0. This implies det(A) = 0 or det(B) = 0. If det(A) ≠ 0, then A is invertible, so B = A<sup>-1</sup>(0) = 0. If det(B) ≠ 0, then B is invertible, so A = (AB)B<sup>-1</sup> = 0. If neither A nor B are null matrices, then both must be singular. If neither det(A) nor det(B) is zero, then AB cannot be a null matrix.
Correct Answer: 1,2,3,4