Question

If A and B are matrices and the columns of AB are independent, show that the columns of B are independent.

Answer #1

If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A
and B are 2x2 matrices, can I use that to show that Det(A)Det(B) =
Det(AB) for any n x n matrix? If so how?

9. Suppose AB is defined and you know all the
entries in both matrices.
a) Find the entry in row i and column j of AB
b) Write column j of AB as a linear combination of
columns of A.
c) Write row i of AB as a liner combination of
rows of B
d) If A is size m×n and
B is size x p , write AB as the
sum of n matrices of...

Prove or disprove: If the columns of B(n×p) ? R are linearly
independent as well as those of A, then so are the columns of AB
(for A(m×n) ? R ).

Problem 30. Show that if two matrices A and B of the same size
have the property that Ab = Bb for every column vector b of the
correct size for multiplication, then A = B.

4.4.3. Suppose A and B are n × n matrices. Prove that, if AB is
invertible, then A and B are both invertible. Do not use
determinants, since we have not seen them yet. Hint: Use Lemma
4.4.4.
Lemma 4.4.4. If A ∈ Mm,n(F) and B ∈ Mn,k(F), then rank(AB) ≤
rank(A) and rank(AB) ≤ rank(B).

Suppose events A,B,C,D are mutually independent. Show that
events AB and CD are independent. Justify each step from the
definition of mutual independence.

Assume A and B are two nonsingular square matrices.
Prove that AB has the same eigenvalue as BA.

Let matrices A,B∈Mn×n(R). Show that if A and B are each similar
to some diagonal matrix, and also have the same eigenvectors (but
not necessarily the same eigenvalues), then AB=BA.

Let A and B be 3x3 matrices with det(A) = 2 and det(B) = -3.
Find
a. det(AB) show all steps.
b. det(2B) show all steps.
c. det(AB^-1) show all steps.
d. det(2AB) show all steps.

What condition should hold in order to multiply two
matrices?
A. The number of columns of the first matrix should be equal to
the rows of the second matrix.
B. The number of rows of the first matrix should be equal to the
columns of the second matrix.
C. None of the above.
D. Two matrices should have the same size.

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