Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant \(\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & \sqrt[2]{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|\) is (A) \(-5 \sqrt{3}(5-\sqrt{6})\) (B) \(-5 \sqrt{3}(5+\sqrt{6})\) (C) \(-5 \sqrt{3}(\sqrt{6}-5)\) (D) None of these

Step-by-Step Solution

Answer

The determinant simplifies to option (B).

Step 1: Set Up the Determinant

Write the given determinant as follows:\[\begin{vmatrix} \sqrt{13}+\sqrt{3} & \sqrt{2}{5} & \sqrt{5} \\sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \3+\sqrt{65} & \sqrt{15} & 5\end{vmatrix}\]

Step 2: Simplify Column 1

Subtract the second row from the first row and the third row from the second row to simplify the calculation. This will make working with easier numbers in the first column.New determinant will become:\[\begin{vmatrix} \sqrt{13}+\sqrt{3} & \sqrt{2}{5} & \sqrt{5} \\sqrt{2} & 5 & \sqrt{10} \3 & \sqrt{15} & 5\end{vmatrix}\] Reduce further if possible.

Step 3: Expand the Determinant

Use the cofactor expansion method to evaluate the determinant. Start with the first row:\[(\sqrt{13}+\sqrt{3}) \cdot \begin{vmatrix}5 & \sqrt{10} \ \sqrt{15} & 5 \end{vmatrix} - (\sqrt{2}{5}) \cdot \begin{vmatrix}\sqrt{2} & \sqrt{10} \ 3 & 5\end{vmatrix} + (\sqrt{5}) \cdot \begin{vmatrix}\sqrt{2} & 5 \ 3 & \sqrt{15}\end{vmatrix}\]

Step 4: Calculate 2x2 Minor Determinants

Evaluate each of the 2x2 minors separately:1. \[\begin{vmatrix} 5 & \sqrt{10} \ \sqrt{15} & 5 \end{vmatrix} = (5)(5) - (\sqrt{10})(\sqrt{15}) = 25 - \sqrt{150}\]2. \[\begin{vmatrix}\sqrt{2} & \sqrt{10} \ 3 & 5\end{vmatrix} = (\sqrt{2})(5) - (\sqrt{10})(3) = 5\sqrt{2} - 3\sqrt{10}\]3. \[\begin{vmatrix}\sqrt{2} & 5 \ 3 & \sqrt{15}\end{vmatrix} = (\sqrt{2})(\sqrt{15}) - (5)(3) = \sqrt{30} - 15\]

Step 5: Substitute and Solve

Substitute the values of the minors back into the determinant equation:\[(\sqrt{13}+\sqrt{3})(25 - \sqrt{150}) - (\sqrt{2}{5})(5\sqrt{2} - 3\sqrt{10}) + (\sqrt{5})(\sqrt{30} - 15)\]Solve step-by-step to find the value of the determinant by carrying out the multiplication and simplification.

Step 6: Simplification and Determine the Correct Answer

Simplify the expression you obtained in Step 5. After simplifying, compare it to the given options (A), (B), (C), and (D). Choose the answer that matches your calculated value.

Key Concepts

Cofactor Expansion

Cofactor expansion is a technique used in evaluating the determinant of a matrix, especially when dealing with matrices larger than 2x2. The method involves breaking down the determinant along a specific row or column into smaller determinants of minors, which are easier to calculate. When performing cofactor expansion, each element of the chosen row or column is multiplied by its cofactor. The cofactor for an element is the determinant of the smaller matrix that remains after removing the row and column of the element, multiplied by \(-1\) raised to the sum of the row and column indices. This sign alternation helps account for the orientation of the matrix.

• For a 3x3 matrix, you can usually choose to expand along the row or column that simplifies your calculations the most.
• Each element in the row or column gets multiplied by its minor determinant, and the signs alternate for the cofactors.

By making strategic choices about where to expand and how to simplify, cofactor expansion can significantly reduce the complexity of determinant calculations.

Matrix Simplification

Matrix simplification plays a crucial role in making determinant evaluation more manageable. Simplification techniques like row and column operations aim to reduce the matrix elements to smaller, more workable numbers, without changing the determinant value. As seen in the example exercise, subtracting one row from another to create zeros or simplifying expressions can be extremely helpful.

• Row operations, such as adding, subtracting, or swapping rows, are useful in transforming the matrix into a simpler form.
• Such operations leverage the properties of determinants and matrices to avoid cumbersome calculations.

The goal of these simplifications is to utilize easier to compute numbers, making the task of expansion and minor calculations more efficient and less error-prone.

Minor Determinants

Minor determinants are an integral part of the cofactor expansion method. A minor is the determinant of a matrix that is one order smaller, created by removing a row and a column from the original matrix. In our exercise, several 2x2 minor matrices were evaluated from the original 3x3 matrix.

• To find a minor, remove the row and column of the element in focus.
• The minor's determinant is the value used in the cofactor calculation for each element of the row or column you are expanding along.

Minor determinants not only simplify calculations by dealing with smaller matrices but also form the building blocks for understanding larger matrix operations. When you master calculating these smaller determinants, evaluating larger matrices becomes a more straightforward task.