Question

For \( j = 1, 2, \dots, 5 \), let \( P_j \) be the matrix of order \( 5 \times 5 \) obtained by replacing the \( j^{th} \) column of the identity matrix of order \( 5 \times 5 \) with the column vector \( v = (5, 4, 3, 2, 1)^T \). Then the determinant of the matrix product \( P_1 P_2 P_3 P_4 P_5 \) is

Hint:

For matrix products, the determinant of the product is the product of the determinants of the individual matrices.

Answer 1

Correct Answer: 120

Step 1: Understanding the matrix product.
Each matrix \( P_j \) is obtained by replacing the \( j^{th} \) column of the identity matrix with the vector \( v \). The product \( P_1 P_2 P_3 P_4 P_5 \) is a product of these modified identity matrices.
Step 2: Compute the determinant.
The determinant of a matrix formed by replacing one column of the identity matrix with a vector can be computed using the properties of determinants. The result for this product is \( 120 \), as the determinant involves the multiplication of the individual determinants of the modified identity matrices.
Step 3: Conclusion.
Thus, the determinant of the matrix product \( P_1 P_2 P_3 P_4 P_5 \) is \( 120 \).