In fact, it can be inferred from "AB reversibility" that "|AB| is not equal to 0" and then "| BA | = | B | * | A | = | A | | B | = | AB | is not equal to 0", so "BA reversibility".
When A and B are not square matrices of the same order, "BA invertibility" cannot be deduced from "AB invertibility".
For example:
A =
[ 1 0 0
0 1 0],
B =
[ 1 0
0 1
0 0],
So AB =
[ 1 0
0 1]
Reversible, but
BA =
[ 1 0 0
0 1 0
0 0 0]
Irreversible
Note: When A and B are square matrices of the same order, the determinant formula "|AB| = |A|*|B|" holds;
When A and B are not square matrices of the same order, for example, in the above example, A is a matrix with 2 rows and 3 columns, and B is a matrix with 3 rows and 2 columns. Although AB and BA are both square matrices, and |AB| and |BA| are meaningful, but A and B are not square matrices, so |A| and |B| are meaningless, so we can see the formula | AB | = | A || B.
References:
Chen: Learning Guide of Linear Algebra ISBN: 97870302 11774, Science Press, March 2008, the first edition1.
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