Correct : c

Definition of a Matrix Inverse:
• If a matrix P is the inverse of another matrix Q (denoted as P = Q^-1), it means that when P and Q are multiplied together, in any order, the result is the identity matrix (I).
• The identity matrix, I, is a square matrix with ones on the main diagonal and zeros elsewhere. When an identity matrix is multiplied by any other matrix of compatible dimensions, it leaves the other matrix unchanged (e.g., A · I = A and I · A = A).
Apply the Definition to the Given Matrices:
• The problem states that "P is the inverse of a matrix Q".
• According to the definition of a matrix inverse, this implies two conditions must be met:     1. P · Q = I
    2. Q · P = I
Given Options:
• A) PQ=I but QP≠I: This is incorrect because for P to be the inverse of Q, both products must equal I.
• B) QP=I but PQ≠I: This is also incorrect for the same reason.
• C) PQ=I and QP=I: This option perfectly aligns with the definition of P being the inverse of Q. When P is the inverse of Q, multiplying them in either order yields the identity matrix.
• D) PQ−QP=I: This is incorrect. Since PQ=I and QP=I, then PQ−QP = I−I = 0 (the zero matrix), not I.
Therefore, the only (c) is the correct statement that satisfies the definition of a matrix inverse is that both PQ and QP must equal the identity matrix.