Let A be a 2*3 matrix, and B be a 3*2 matrix such that |AB|=4 Find the value of |BA|
The title is a big hint: focus on the rank of the matrices.
The rank of a matrix cannot be larger than the smaller dimension of the matrix. Then rank(A)<=2 and rank(B)<=2.
The rank of a product of matrices cannot be greater than the rank of any one of the multiplicands. Then rank(BA)<=2.
From the Invertible Matrix Theorem, the rank of an invertible matrix of size n is equal to n and the determinant of the matrix is nonzero. From which we can state if a matrix of size n has a rank less than n it is then uninvertible and has a determinant equal to 0. BA is a size 3 matrix with a rank of at most 2, therefore |BA| = 0.
Solution