Consider the following pairs:

a. Square matrix has two identical rows 1. Determinant = 1

b. 3 x 3 Identity matrix

 2. Determinant = 4

c. 3 X 3 diagonal matrix with diagonal entries 1, 2 and 3 3. Determinant = 0

d. P and Q are two 3 X 3 matrices with det(P) = 2 and det (Q) = 4. Then the determinant of PQT is 4. Determinant = 6

 5. Determinant = 8

Which of the pairs given above is/are correctly matched?

To determine the correct matches, we apply the fundamental properties of determinants for each case:a — 3 (Square matrix with two identical rows)Property: If any two rows (or columns) of a square matrix are identical, the determinant of that matrix is zero.Result: Determinant = 0. (Matches a-3)b — 1 (3 x 3 Identity matrix)Property: The determinant of an identity matrix ($I$) of any order is always one.Result: Determinant = 1. (Matches b-1)c — 4 (3 x 3 Diagonal matrix with entries 1, 2, 3)Property: The determinant of a diagonal matrix is the product of its diagonal elements.Calculation: $1 \times 2 \times 3 = 6$.Result: Determinant = 6. (Matches c-4)d — 5 (Matrices P and Q with det(P)=2 and det(Q)=4)Property 1: $\det(AB) = \det(A) \times \det(B)$Property 2: The determinant of a matrix is equal to the determinant of its transpose, i.e., $\det(Q) = \det(Q^T)$.Calculation: $\det(PQ^T) = \det(P) \times \det(Q^T) = 2 \times 4 = 8$.Result: Determinant = 8. (Matches d-5)