If A=| 1 logx y logx z logy x 1 logy z logz x logz y 1 |, then

step 1 So in the given question we are given a matrix a which is given as a is equal to 1 log with base

If A=| 1 logx y logx z logy x 1 logy z...

If $A=\left|\begin{array}{ccc}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|$, then $|A|$ is equal to
(a) 0
(b) $\log \mathrm{xyz}$
(c) 1
(d) $\log _{y} x+\log _{2} y+\log _{x} z$

Key Concepts

Linear Dependence in Matrices

A set of matrix rows (or columns) can be linearly dependent if one row can be expressed as a combination of the others. This situation results in a determinant of zero. Recognizing such dependencies, especially when the entries are linked via logarithmic identities, is key in evaluating determinants efficiently.

Change of Base Formula

This formula allows the conversion of logarithms from one base to another, which is particularly useful in situations where a matrix contains logarithms with varying bases. The formula, log_b(a) = log_k(a) / log_k(b), facilitates comparing and manipulating logarithmic expressions.

Matrix Determinant

The determinant is a scalar value derived from a square matrix that provides key insights into the matrix's properties, such as invertibility. It is computed using combinations of the matrix's entries and can be zero if the rows (or columns) are linearly dependent.

Properties of Logarithms

Logarithms follow several fundamental rules, such as the product, quotient, and power rules, which simplify expressions involving logs. These properties are essential when working with matrices that incorporate logarithmic terms, as they allow for the conversion and simplification of different bases.

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