Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
<div>22. $D = \begin{vmatrix} 10^4 + 2 & 10^7 + 3 & 10^8 + 8 \\ 10^9 + 9 & 10^2 + 8 & 10^3 - 4 \\ 10^3 - 5 & 10^8 + b & 10^6 + a \end{vmatrix}$ where $a, b$, both $\in \{1,2,3,4,5,6,7,8,9\}$<br>Number of ordered pairs $(a, b)$ such that $D = 2n + 1, n \in Z$ is</div>
(A) 36
(B) 20
(C) 16
(D) 45
Step-by-Step Solution
Key Concept: The determinant D is evaluated modulo 2. Since 10^k is even for k >= 1, the determinant simplifies significantly when considering parity.
The determinant D modulo 2 is equivalent to the determinant of the matrix where each entry is replaced by its parity. Since 10^k is even for k >= 1, the matrix modulo 2 becomes: <br> D \equiv |0+0, 0+1, 0+0; 0+1, 0+0, 0+0; 0+1, 0+0, 0+a| (mod 2) <br> D \equiv |0, 1, 0; 1, 0, 0; 1, 0, a| (mod 2) <br> Expanding along the first row: 0 - 1(a - 0) + 0 = -a \equiv a (mod 2). <br> For D to be odd (2n+1), a must be odd. The possible values for a are {1, 3, 5, 7, 9} (5 values). The value of b does not affect the parity of the determinant, so b can be any of the 9 values {1, 2, ..., 9}. Total pairs = 5 * 9 = 45.
Correct Answer: 4