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?

a matches with 3 (Determinant = 0): A fundamental property of matrices is that if any two rows or columns are identical, the determinant of that matrix is automatically zero.

b matches with 1 (Determinant = 1): An identity matrix of any size always has a determinant equal to one because it represents a neutral transformation that does not scale volume.

c matches with 4 (Determinant = 6): For a diagonal matrix, you can find the determinant simply by multiplying all the numbers along the main diagonal together. Multiplying one, two, and three gives you six.

d matches with 5 (Determinant = 8): The determinant of a product of matrices is equal to the product of their individual determinants. Furthermore, taking the transpose of a matrix does not change its determinant. Therefore, to find the determinant of the product of matrix P, matrix Q, and the transpose of matrix Q, you multiply the determinant of P (which is two) by the determinant of Q (which is four). Multiplying two by four gives you eight.