11. Prove one of the following: a. Let A and B be square matrices. If det(AB)...

Question

Question

math algebra

Answer 1

Answer #1

(a)

Note that, det(AB) ≠ 0 means det(A) ≠ 0 and at the same time det(B) ≠ 0

Now det(B) ≠ 0 implies, B is invertible.

.

(b)

Since A is a n×n matrix and the equation Ax = 0 has a non-trivial solution then the row reduced echelon form of A has (n - 1) pivot columns.

This means that, Rank(A) = n - 1

Now as, n - 1 < n, so we have,

Rank(A) < n

Answer 2

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