If a, b, c are three integers in A.P., lying between 1 and 9...

If $a, b, c$ are three integers in A.P., lying between 1 and 9 and $a 31, b 41$ and $c 51$ are three digit numbers, then the determinant
$$
\left|\begin{array}{ccc}
3 & 4 & 5 \\
a 31 & b 41 & c 51 \\
a & b & c
\end{array}\right| \text { equals }
$$
(a) $a+b+c$
(b) $3 b$
(c) $=1$
(d) 0

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Key Concepts

Arithmetic Progressions

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant value, known as the common difference, to the previous term. A key property is that the middle term in a three-term arithmetic progression is the average of the first and third terms, a fact that is often used to simplify calculations or reveal hidden relationships in problems.

Determinants and Their Properties

The determinant is a scalar value calculated from a square matrix that provides insight into many properties of the matrix, such as the matrix's invertibility and the linear independence of its rows or columns. Specifically, if the rows (or columns) of a matrix are linearly dependent, the determinant of the matrix will be zero, which is an important property that often simplifies the evaluation of complex determinants.

Linear Dependence in Matrices

Linear dependence among rows or columns of a matrix means that one row (or column) can be written as a linear combination of others. This concept is critical when computing determinants because if such a dependency exists, the determinant immediately becomes zero, streamlining problem-solving in linear algebra contexts.

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