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Q. Let A be a 2x3 matrix whereas B be a 3x2 matrix. If det(AB) = 4, then the value of det(BA) is ? (Ans: 0) I tried this problem by taking 2 to 3 examples which satisfy the conditions in the questions and got the answer. But I am not finding a suitable method or any theorem that can be applied.

Q. Let A be a 2x3 matrix whereas B be a 3x2 matrix. If det(AB) = 4, then the value of det(BA) is ?

(Ans: 0)

 

I tried this problem by taking 2 to 3 examples which satisfy the conditions in the questions and got the answer. But I am not finding a suitable method or any theorem that can be applied.

Grade:12

2 Answers

Harsh Patodia IIT Roorkee
askIITians Faculty 907 Points
7 years ago
As such there is no theorem for this. Its best to take examples.
mycroft holmes
272 Points
7 years ago
Let
 X = \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}
 
Since A is a 2X3 matrix, the equation AX = 0 has a non-trivial solution. and hence the equation (BA) X = 0 has a non-trivial solution, with BA being a 3X3 matrix.
 
Hence, BA is not invertible, i.e. det (BA) = 0

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