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Matrix rank? (Posted on 2019-03-24) Difficulty: 3 of 5
Let A be a 2*3 matrix, and B be a 3*2 matrix such that |AB|=4 Find the value of |BA|

No Solution Yet Submitted by Danish Ahmed Khan    
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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.

  Posted by Brian Smith on 2019-03-24 13:06:55
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