Question

Prove that if the product AB of two n × n matrices is invertible, then both A and B are invertible. Even if you know about determinants, do not use them, we did not cover them yet. Hint: use previous 2 problems.

Answer 1

Answer:

Step-by-step explanation:

Given two matrices A and B

The product of the matrix A and B is AB

And we know that AB is an "n×n" matrix

We can defined invertible by the rank of the matrix, then, RANK of a matrix is equal to the maximum number of linearly independent columns or rows in the matrix

Also, the rank of an "n×n" matrix is at most "n", an "n×n" having a rank "n" is invertible.

Since, AB is an "n×n" matrix,

AB will be invertible if and only if RANK(AB)=n, and it has a pivots in every rows and columns, so that it ranks will be n,

NOTE: If matrix A is an "n×n"

, if matrix B is an "n×n"

Then, the rank of A cannot be more than "n" and the rank of B cannot be more than "n".

Therefore, Both A and B have rank "n" and they have a pivots in every row and column, so they are both invertible.