If A1 B1 C1, A2 B2 C2 and A3 B3 C3 are three three-digit...

If $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$ and $A_{3} B_{3} C_{3}$ are three three-digit numbers, each of which is divisible by $k$, then $\Delta=\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|$ is (A) divisible by $k$ (B) divisible by $k^{2}$ (C) divisible by $2 k$ (D) None of these

If $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$ and $A_{3} B_{3} C_{3}$ are three three-digit numbers, each of which is divisible by $k$, then
$\Delta=\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|$ is
(A) divisible by $k$
(B) divisible by $k^{2}$
(C) divisible by $2 k$
(D) None of these

Key Concepts

Determinants in Linear Algebra

The determinant is a fundamental scalar value associated with a square matrix that encapsulates various properties of the linear transformation represented by the matrix. Key properties include its behavior under row operations and its multiplicative nature. In problems that mix algebra with number theory, properties of determinants can be exploited to determine the divisibility of the determinant based on the divisibility properties of the entries, as is examined in this question.

Divisibility and Modular Arithmetic

This concept deals with determining when one number is an exact multiple of another, an idea that is formalized using the language of modular arithmetic. In many problems, such as the one considered, the properties of numbers modulo a given integer are analyzed to deduce information about larger algebraic expressions, including determinants. Understanding divisibility enables one to trace how factors, like k in this case, influence the arithmetic structure of composite expressions.

Representation of Numbers in Positional Systems

This concept involves expressing numbers based on the place value of digits, such as writing a three?digit number as 100A + 10B + C. Recognizing this representation is important when applying divisibility rules since the positional weights (like 100, 10, and 1) contribute to the overall divisibility properties. By breaking down numbers in this form, one can analyze how divisibility conditions for the entire number translate into relations between its digits and subsequently affect related algebraic expressions like determinants.

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