Problem 69 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x_{1} \neq 0, x_{2} \neq 0, x_{3} \neq 0\), then the determinant \(\left|\begin{array}{ccc}x_{1}+a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\\ a_{2} b_{1} & x_{2}+a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & x_{3}+a_{3} b_{3}\end{array}\right|\) is equal to (A) \(x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)\) (B) \(-x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)\) (C) \(x_{1} x_{2} x_{3}\left(1-\frac{a_{1} b_{1}}{x_{1}}-\frac{a_{2} b_{2}}{x_{2}}-\frac{a_{3} b_{3}}{x_{3}}\right)\) (D) None of these

Step-by-Step Solution

Answer

The determinant is Option (A).

Step 1: Identify Mathematical Form

The given matrix is a 3x3 matrix with elements expressed as combinations of variables and constants.

Step 2: Apply Determinant Formula

The determinant of a 3x3 matrix \( \left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right| \) is calculated using the formula \( a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 3: Substitute Values

Substituting the given values into the determinant formula:\( (x_1 + a_1b_1)((x_2 + a_2b_2)(x_3 + a_3b_3) - (a_2b_3)(a_3b_2)) - (a_1b_2)((a_2b_1)(x_3 + a_3b_3) - (a_2b_3)(a_3b_1)) + (a_1b_3)((a_2b_1)(a_3b_2) - (x_2 + a_2b_2)(a_3b_1)) \).

Step 4: Simplify Each Term

For the first term, simplify:\( (x_2 + a_2b_2)(x_3 + a_3b_3) - (a_2b_3)(a_3b_2) \).Upon expansion, you get:\( x_2x_3 + a_2b_2x_3 + a_3b_3x_2 + a_2b_2a_3b_3 - a_2b_3a_3b_2 \).

Step 5: Expand Entire Expression

Substitute the simplified term back into the original expression and expand. After expanding, collect similar terms.

Step 6: Factor Out Products

Factor out the products that include \(x_1, x_2, x_3\). You will commonly recognize terms in the format \(x_1x_2x_3(1 + \text{other terms})\).

Step 7: Recognize the Determinant Structure

Recognize that the matrix is constructed like \(A + uv^T\), where \(A\) is a diagonal matrix. Using the formula for determinants of such matrices, calculate \((1 + u^Tv)\times\text{det}(A)\).

Step 8: Determine the Final Result

After calculating the determinant using the steps outlined, the answer matches the expression:\(x_1 x_2 x_3 \left(1 + \frac{a_1 b_1}{x_1} + \frac{a_2 b_2}{x_2} + \frac{a_3 b_3}{x_3}\right)\). This corresponds to Option (A).

Key Concepts

Matrix Algebra

Matrix Algebra is a fundamental identity in the field of linear algebra. It involves operations like addition, subtraction, multiplication, and finding determinants. In this exercise, understanding a 3x3 matrix is crucial.
Matrices are essentially arrays of numbers, organized in rows and columns.
In the context of a 3x3 matrix, you have nine elements, laid out in three rows and three columns.

• Each number in the matrix is referred to as an element and has a unique position based on its row and column.
• Matrices help solve systems of linear equations and perform many complex transformations.
• Understanding how to systematically navigate through a matrix is key to solving determinant calculations.

To better visualize it, imagine each element's position like on a tic-tac-toe grid.
As you delve deeper into solving determinants or other matrix operations, locating and understanding these positions become essential.

Determinant Calculation

Determinant Calculation for a 3x3 matrix is an important task in mathematics, particularly in linear algebra.
To calculate the determinant for any 3x3 matrix, you must follow a specific formula which involves the elements of the matrix.
For a matrix:e.g., \[\left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right|\]the determinant is calculated as:\[a(ei - fh) - b(di - fg) + c(dh - eg)\].

• Each term in this formula represents a unique combination of matrix elements multiplied together, following the rules of matrix algebra.
• When computing such determinants, always be mindful of each sign and operation as they significantly impact the final result.
• Systems of equations, vector space transformations, and understanding the independence of vectors are just a few applications of determinants.

The determinant can tell a lot about a matrix, such as whether the matrix is invertible or not.
Hence, the computation of the determinant is more than a mere calculation but a gateway to understanding deeper mathematical insights.

Properties of Determinants

The Properties of Determinants are both captivating and essential, providing insights into matrix behavior.
They streamline computations and help predict results without exhaustive calculations.

• One key property is that switching two rows or columns of a matrix multiplies the determinant by -1. This reflects symmetry and balance inherent in matrices.
• If any row or column contains only zeros, the determinant of that matrix is zero. This indicates a dependency or lack of dimensionality in the system described by the matrix.
• The determinant of a matrix product, such as \(AB\), is the product of their determinants: \(\det(A) \times \det(B)\). This is crucial in matrix decomposition where matrices are broken down into simpler forms.
• Adding a multiple of one row to another does not change the determinant's value. This property comes into play during operations like Gaussian elimination which simplifies matrices while keeping determinant values reliable.

These properties make manipulation of matrices straightforward and valuable, allowing for efficient simplification and analysis of matrix-related problems.