Answer to Question #24412 in Linear Algebra for [e-mail]

Question

1. (2 Points) Explain clearly why the solution to the homogeneous system Ax = 0 with a nonsingular
coecient matrix is x = 0.

2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.

3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.

Answer 1

1.
If A is nonsingular then detA <>0 and there is A^(-1). So x=A^(-1)0=0

2.
If there is some F that DF=FD=I then F have to be too diagonal and
F = diag(f1,f2,.....,fn).
Then we have that f_i * d_i = 1.
So, D is invertible iff each d_i is invertible, and then
D^(-1)=diag{d1^(-1), ... , dn^(-1)}.

3.
It is well known that if a is m by n matrix then rank(A) <= min{m,n}.
In our case we have that rank(A)<=n, rank(B)<=n, also for any matrices we
know that rank(AB)<=min{rank(A),rank(B)}.
We have that AB will be m by m matrix and thus it will be invertible iff
det(AB)<>0 iff rank(AB)=m, but
rank(AB)<=min{rank(A),rank(B)}<=n<m
so, AB is not invertible.