Problem 42 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of the determinant \(\left|\begin{array}{ccc}\sqrt{x}+\sqrt{y} & 2 \sqrt{z} & \sqrt{z} \\ \sqrt{y z}+\sqrt{2 x} & z & \sqrt{2 z} \\ y+\sqrt{x z} & \sqrt{y z} & z\end{array}\right|\) where \(x, y, z\) are positive real numbers, is (A) \(z(\sqrt{2} y-z \sqrt{y)}\) (B) \(y(\sqrt{2} z-y \sqrt{z)}\) (C) \(x(\sqrt{2} y-z \sqrt{y})\) (D) None of these

Step-by-Step Solution

Answer

Option (C) is correct: \(x(\sqrt{2}y - z\sqrt{y})\).

Step 1: Simplify the Matrix

Start by simplifying the given matrix with elements \(a_{ij}\), where the elements are functions of \(x, y,\) and \(z\). The matrix is: \[ \begin{bmatrix} \sqrt{x} + \sqrt{y} & 2\sqrt{z} & \sqrt{z} \\sqrt{yz} + \sqrt{2x} & z & \sqrt{2z} \y + \sqrt{xz} & \sqrt{yz} & z \\end{bmatrix} \] Verify that all terms are in their simplest form.

Step 2: Recognize Patterns or Strategies

The matrix does not easily simplify into known forms of determinants, so evaluate via row or column operations if possible. Notice that the problem might involve strategic simplifications by exploiting symmetries or strategic row and column operations.

Step 3: Attempt Row and Column Operations

Perform column operations to simplify calculation. For example, try performing operations where you subtract multiples of one column from another, such as C2 = C2 - 2C3, to see if simpler forms or cancelation occurs.

Step 4: Calculate Determinant by Standard Method

Use the standard determinant calculation method for a 3x3 matrix. For matrix \(A\) with \(a, b, c\) as the first row, \(d, e, f\) as the second row, \(g, h, i\) as the third row, the determinant \(|A|\) is \[ |A| = a(ei-fh) - b(di-fg) + c(dh-eg) \] Apply this formula to compute \(|A|\) based on recent operations if possible.

Step 5: Compare with Given Options

After determining the simplified determinant expression, compare your final answer with the given choices. Identify which option correctly matches the expression derived through calculation.

Key Concepts

Row and Column Operations

When working with matrices, row and column operations are powerful techniques used to simplify the calculation of determinants or to transform the matrix into a more manageable form. These operations include swapping rows or columns, multiplying a row or column by a non-zero scalar, and adding or subtracting multiples of one row or column to another row or column. Each of these operations affects the determinant in specific ways.

There are some key points to keep in mind:

• Swapping two rows or columns changes the sign of the determinant.
• Multiplying a row or column by a scalar multiplies the determinant by the same scalar.
• Adding or subtracting a multiple of one row or column to another does not change the determinant.

In the original solution, column operations were used to simplify the determinant calculation. For instance, subtracting multiples of one column from another can sometimes lead to zeros in the matrix, making the determinant calculation straightforward.

3x3 Matrix

A 3x3 matrix consists of three rows and three columns and can be represented in the following form:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]The determinant of a 3x3 matrix is a scalar value that provides important information about the matrix properties, such as whether the matrix is invertible. Calculation involves an expansion method that uses minors and cofactors.

For the determinant \(|A|\) of a 3x3 matrix to be computed efficiently, it's essential to follow this formula:\[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\]This method of calculation shows that each term involves multiplying the element by the determinant of the 2x2 matrix formed by deleting the row and column of that element. The determinant plays a pivotal role in various applications, including solving systems of linear equations, eigenvalue problems, and transformations in three-dimensional space.

Calculus Problem

While the problem at hand is fundamentally an algebraic one involving determinants, it showcases how calculus principles occasionally intersect with linear algebra ideas. In this exercise, the square roots within the matrix elements suggest functions derived from possible calculus operations, like integrals or derivatives. Specifically, it shows the presence of expressions that need manipulation and simplification, akin to calculus-based problem-solving.

Further, when working on determinants, understanding the structure and behavior of functions within matrix elements is crucial. For calculus students, recognizing patterns of simplification in algebraic expressions is beneficial, as this may foreshadow integrals or derivatives requiring similar strategies for simplification. In this determinant problem, identifying and performing suitable column operations embodies a typical calculus approach: modifying expressions for simpler, more interpretable forms. By practicing these skills within matrix operations, students also hone the critical thinking needed to tackle complex calculus problems effectively.