Correct : c

To determine the correct relationship between a matrix and its inverse, let us recall the fundamental definition of an inverse matrix.
Definition of a Matrix Inverse:
• A square matrix A is said to be invertible if there exists a square matrix B of the same size such that:
    A × B = I
    and
    B × A = I
where I is the identity matrix of the same dimension.

• The matrix B is called the inverse of A, denoted as A^-1. Conversely, A is also the inverse of B.

Apply to the Given Problem:
• The problem states that "The matrix P is the inverse of a matrix Q."
• According to the definition, if P is the inverse of Q, then multiplying P and Q in any order must yield the identity matrix I.

• Therefore, we must have:
    P × Q = I
    and
    Q × P = I

Evaluate the Options:
Let's check each given option against this definition:
• A) PQ=I but QP≠I: This is incorrect because both products must be I.
• B) QP=I but PQ≠I: This is incorrect for the same reason.
• C) PQ=I and QP=I: This option perfectly matches the definition of matrix inverse.
• D) PQ−QP=I: Since PQ=I and QP=I, then PQ−QP = I−I = 0 (the zero matrix), not I. So, this is incorrect.

Based on the definition, the only correct option is that both PQ and QP must equal the identity matrix I.