If | 1+a 1 1 1 1+b 1 1 1 1+c |=abc, then 1 / a+1 / b+1 / c

step 1 So in the given question we are told that there is a matrix which has the determinant. The matri

If | 1+a 1 1 1 1+b 1 1 1 ...

If $\left|\begin{array}{ccc}1+\mathrm{a} & 1 & 1 \\ 1 & 1+\mathrm{b} & 1 \\ 1 & 1 & 1+\mathrm{c}\end{array}\right|=\mathrm{abc}$, then $1 / \mathrm{a}+1 / \mathrm{b}+1 / \mathrm{c}$ equals
(a) -1
(b) 1
(c) 0
(d) 2

Key Concepts

Reciprocal Sums and Their Relationship to Products

In algebra, the sum of reciprocals of variables is often expressed as a ratio of the sum of pairwise products to the overall product. Recognizing this relationship enables one to evaluate reciprocal sums efficiently when a condition like equality to a product is given. This concept is particularly useful when the algebraic manipulation of a determinant leads to a relation among the variables that can be directly applied to calculate the sum of reciprocals.

Determinant Evaluation

Determinants are a fundamental concept in linear algebra that assign a scalar value to a square matrix. For a 3×3 matrix, the determinant can be calculated using cofactor expansion or by applying row operations. This process transforms the matrix into a simpler form, making the calculation easier and revealing important properties such as linear dependence or the relationship between the matrix's elements.

Row Operations in Determinants

Row operations, such as subtracting one row from another, are useful in simplifying the evaluation of determinants. They allow us to create zeros in strategic locations within the matrix, which reduces the complexity of the calculation. Importantly, specific row operations, like subtracting a multiple of one row from another, do not change the value of the determinant, thereby preserving the key relationships among the matrix elements.

Algebraic Manipulation and Factorization

Once the determinant is expressed in terms of the variables, algebraic manipulation and factorization become essential. These processes help in reorganizing the expression to reveal hidden relationships among the variables. In problems where a determinant is equated to a product involving variables, factorization can lead to conditions that simplify the evaluation of other expressions, such as sums or reciprocals.

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