Problem 35 - Algebra for JEE ADVANCED

Question

If \(\left|\begin{array}{lll}x & 3 & 6 \\ 3 & 6 & x \\ 6 & x & 3\end{array}\right|=\left|\begin{array}{lll}2 & x & 7 \\ x & 7 & 2 \\ 7 & 2 & x\end{array}\right|=\left|\begin{array}{ccc}4 & 5 & x \\ 5 & x & 4 \\ x & 4 & 5\end{array}\right|=0\), then \(\cdot x^{*}\) is equal to a. 0 b. \(-9\) c. 3 d. none of these

Step-by-Step Solution

Answer

The value of \( x \) is -9.

Step 1: Understand the Problem

We are given three distinct 3x3 matrices whose determinants are all equal to zero. We need to find the value of \( x \) that satisfies this condition.

Step 2: Matrix Determinant Equals Zero

If the determinant of a matrix is zero, it implies that the rows or columns of the matrix are linearly dependent. This leads to polynomial equations in \( x \) for each matrix when we set their determinants to zero.

Step 3: Calculate Determinant of First Matrix

The first matrix is \( \begin{bmatrix} x & 3 & 6 \ 3 & 6 & x \ 6 & x & 3 \end{bmatrix} \). Calculate its determinant: \( x(6\cdot3 - x^2) - 3(3\cdot3 - 6x) + 6(x\cdot6 - x^2) \). Set this determinant to zero and simplify the equation.

Step 4: Calculate Determinant of Second Matrix

The second matrix is \( \begin{bmatrix} 2 & x & 7 \ x & 7 & 2 \ 7 & 2 & x \end{bmatrix} \). Similarly, calculate its determinant and set it to zero. This will give another polynomial equation in \( x \).

Step 5: Calculate Determinant of Third Matrix

Finally, the third matrix is \( \begin{bmatrix} 4 & 5 & x \ 5 & x & 4 \ x & 4 & 5 \end{bmatrix} \). Calculate this matrix's determinant, set it to zero, and obtain a third polynomial equation.

Step 6: Solve the System of Polynomial Equations

The system comprises three polynomial equations from the determinants set to zero. Solve the system to find the possible values for \( x \). This involves finding common roots of all three equations.

Step 7: Verify and Select the Correct Option

After solving the equations, verify the found \( x \) values against the given options 0, -9, 3, and other options if any. Select the option that matches one of the roots.

Key Concepts

Linear Dependence

In the context of matrices, linear dependence refers to a situation where one or more rows (or columns) of a matrix can be expressed as a linear combination of other rows (or columns). This means they lie in the same lower-dimensional space and do not add any new dimension to the space formed by the matrix.

When dealing with determinants, if the determinant of a matrix is zero, it indicates that the rows or columns of the matrix are linearly dependent. In simpler terms, the matrix is not full rank, and it does not have an inverse.

For example, in a 3x3 matrix, linear dependence of its rows implies that you can find constants, not all zero, such that they can be added together to form the zero row vector. Thus, solving the determinants of matrices often involves checking for linear dependence.

Polynomial Equations

Polynomial equations are mathematical expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative power operations of variables.

When solving for the determinant of a matrix where one or more terms are variables, such as 'x', the expression for the determinant will form a polynomial equation. In this context, polynomial equations arise as each determinant of a 3x3 matrix leads to a cubic or quadratic polynomial equation in terms of 'x'.

• The degree of the polynomial is determined by the largest power of 'x' present in the determinant calculation.
• Setting these polynomial equations to zero helps find the potential 'x' values that make the matrix determinant zero and thus exhibit linear dependence.

Solving these polynomial equations provides crucial information on what values the variable needs to take in order for the determinant condition (like being zero) to be met.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. It is often written in the form: \[ \begin{bmatrix}a & b & c \ d & e & f \ g & h & i \end{bmatrix}\] Each element in the matrix is placed in a specific position according to row and column indices.

The determinant of a 3x3 matrix provides critical insights, such as:

• Indicating invertibility: If the determinant is non-zero, the matrix is invertible. If it is zero, the matrix is not invertible.
• Relating to volume: In geometry, the determinant gives the volume scaling factor of the transformation described by the matrix.

For linear algebra applications, finding the determinant is key to solving systems of equations, determining dependencies and understanding transformations.

System of Equations

A system of equations in mathematics involves finding values for variables that satisfy multiple equations simultaneously. For linear algebra, specifically referring to systems with variables from matrices, these equations can often result from determinant problems.

When dealing with matrices, if you have multiple determinant equations set to zero, it results in a system of polynomial equations.

To solve such systems:

• Each equation arising from a determinant provides a separate condition the variable must satisfy.
• Solving the system means finding variable values that work for all equations at once - these are common roots of the polynomial equations.

Successfully solving a system of equations involving multiple matrices' determinants helps understand which conditions allow the matrices to exhibit linear dependence, adding to the toolkit used for tackling algebraic and geometric challenges.