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In: Advanced Math

Show that the inverse of an invertible matrix A is unique. That is, suppose that B...

Show that the inverse of an invertible matrix A is unique. That is, suppose that B is any matrix such that AB = BA = I. Then show that B = A−1 .

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Question which I saw has following solution :

Question : is it possible to system of linear equations having n equations and n unknowns (variables) to have a unique solution if the reduced row-echelon form of the associated augmented matrix has a row of zeros?

Solution : Unique solution is not possible in that case.

If A is the coefficients matrix of system of linear equations and M it's augmented matrix and 'n' be the no. of unknowns :

1) Rank(A) ≠ Rank(M) -> No solution

2) Rank (A) = Rank(M) < n -> infinite solutions

3) Rank(A) = Rank(M) = n. -> unique solution

So if in augmented matrix, there is one zero row, that means rank(A) = rank(M) but now rank will be (n-1) which is less than no of unknowns. So,there are infinite solutions as there will be one free variable. (less equations and more unknowns)

Colby Messinger answered 1 year ago
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