Consider the following pairs c. 3×3 diagonal matrix with diagonal entries 1, 2 and 3

Question

Consider the following pairs:

a. Square matrix has two identical rows

b. 3×3 Identity matrix

c. 3×3 diagonal matrix with diagonal entries 1, 2 and 3

d. P and Q are two 3×3 matrices with det(P) = 2 and det(Q) = 4. Determinant of PQ^T is

Match with:

1. Determinant = 1

2. Determinant = 4

3. Determinant = 0

4. Determinant = 6

5. Determinant = 8

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

1. A. a-3, b-1, c-4, d-5 (Correct answer)
2. B. a-3, b-2, c-5, d-1
3. C. a-4, b-1, c-2, d-5
4. D. a-1, b-2, c-3, d-4

Correct Answer

Option A — a-3, b-1, c-4, d-5

Detailed Solution & Explanation

The correct answer is a-3, b-1, c-4, d-5.

Key Points

• a. Identical rows → det=0 (option 3) ✓. b. 3×3 Identity matrix → det=1 (option 1) ✓. c. Diagonal entries 1,2,3 → det=1×2×3=6 (option 4) ✓. d. det(PQ^T)=det(P)×det(Q)=2×4=8 (option 5) ✓. Match: a-3, b-1, c-4, d-5.

Additional Information

• Determinant facts to carry: a matrix with two identical rows (or columns) has determinant 0; the identity matrix has determinant 1; a diagonal or triangular matrix has determinant equal to the product of its diagonal entries.
• Also useful: |AB| = |A||B|, |Aᵀ| = |A|, and multiplying a single row by k multiplies the determinant by k.
• For an n × n matrix, |kA| = kⁿ|A| — the power of n catches people out.
• A matrix is invertible (non-singular) exactly when its determinant is non-zero.