Problem 46 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x \neq 0, y \neq 0, z \neq 0\) and \(\left|\begin{array}{ccc}1+x & 1 & 1 \\\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0\), then \(x^{-1}+y^{-1}+z^{-1}\) is equal to (A) \(-1\) (B) \(-2\) (C) \(-3\) (D) None of these

Step-by-Step Solution

Answer

None of the given options match; the value is \(-\frac{11}{6}\).

Step 1: Write the Determinant

Identify the components of the given determinant: \( \begin{vmatrix} 1+x & 1 & 1 \ 1+y & 1+2y & 1 \ 1+z & 1+z & 1+3z \end{vmatrix} = 0 \).

Step 2: Simplify Using Row Operations

Apply row operations to simplify the determinant. Subtract the first row from the second and third rows to make calculations simpler: \[ R_2 \rightarrow R_2 - R_1 \] and \[ R_3 \rightarrow R_3 - R_1 \]. This results in:\[\begin{vmatrix}1+x & 1 & 1 \y-x & y & 0 \z-x & z & 2z \end{vmatrix}= 0.\]

Step 3: Expand the Determinant

Compute the determinant by expanding along the first row:\[(1+x)(y \cdot 2z - 0) - (1)(y(z-x)) + (1)(y(z-x)) = 0.\]Simplify to obtain \((1+x)yz = 0\).

Step 4: Analyze the Simplified Expression

Since none of \(x, y,\) or \(z\) are zero, \((1+x)yz = 0\) implies \(1+x = 0\). This leads to \(x = -1\).

Step 5: Solve for Individual Values

As we know \(x = -1\), substitute back into the matrix and recompute or logically deduce: for the equation to hold true, similar deductions are applied to solve for \(y = -2\) and \(z = -3\).

Step 6: Calculate Required Expression

Compute \(x^{-1} + y^{-1} + z^{-1}\):\(\frac{-1}{1} + \frac{-1}{2} + \frac{-1}{3} = -1 - \frac{1}{2} - \frac{1}{3}.\)Continue calculation:\(= -\frac{6}{6} - \frac{3}{6} - \frac{2}{6} = -\frac{11}{6}.\)

Step 7: Validate Options and Conclude

Compare this result \(-\frac{11}{6}\) with the given options. None of the options match the calculated value.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics involving matrices, which are arrays of numbers handled using algebraic operations. Matrices are crucial in solving systems of equations and in transforming geometric data. In this context, a matrix determinant is a special numerical value that is calculated from its elements.
To understand determinants, visualize them as providing a "volume" or "scaling factor" information about a transformation represented by the matrix.
Calculating determinants through matrix algebra helps determine properties like the invertibility of a matrix. If the determinant is zero, as in the given exercise, the matrix is singular, which means it does not have an inverse, indicating dependencies among its rows or columns.

Row Operations

In matrix algebra, row operations are a vital tool for simplifying complex problems, especially when calculating determinants. Row operations allow us to manipulate the rows of a matrix to achieve a simpler equivalent form, making calculations easier.
There are primarily three types of row operations:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a scalar multiple of one row from another row

In the given problem, row operations were used to simplify the original matrix into a form that made it easier to calculate the determinant. Subtracting the first row from the second and third rows was essential to reach a simpler expression, unveiling dependencies among variables.

Inverse Operations

Inverse operations are key to solving matrix equations and finding solutions to systems of linear equations. For a matrix to have an inverse, its determinant must be non-zero.
In the given problem, the determinant is zero, meaning the matrix does not have an inverse. This indicates that there is a linear dependency among the matrix rows, which prevents the inverse from existing. Solving the problem involves simplifying the determinant's expression and using the resulting equations to solve for the variables.
Recognizing when a zero determinant renders a matrix non-invertible is important to understand system constraints, leading to adjustments in solving the problem by focusing on logical deductions from simplified expressions.