Problem 20 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(x^{2}+y^{2}+z^{2}=1\), then what is the value of \(\left|\begin{array}{ccc}1 & z & -y \\ -z & 1 & x \\ y & -x & 1\end{array}\right|\) is: (a) 0 (b) 1 (c) 2 (d) \(2-2 x y z\)

Step-by-Step Solution

Answer

The value of the determinant is \( 2 - 2xyz \), option (d).

Step 1: Recognize the Determinant Formula

We are given a 3x3 matrix and need to calculate its determinant. For a matrix \( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant \( |A| \) is calculated as \( a(ei-fh) - b(di-fg) + c(dh-eg) \).

Step 2: Identify Matrix Elements

The elements of the given matrix \( \begin{pmatrix} 1 & z & -y \ -z & 1 & x \ y & -x & 1 \end{pmatrix} \) are identified as follows: \( a=1, b=z, c=-y, d=-z, e=1, f=x, g=y, h=-x, i=1 \).

Step 3: Apply the Determinant Formula

Substitute the matrix elements into the determinant formula:\[1(1 \cdot 1 - x \cdot (-x)) - z((-z \cdot 1 - x \cdot y)) + (-y)((-z)(-x) - y) \]

Step 4: Simplify the First Term

Calculate \( 1(1+x^2) = 1 + x^2 \). This simplifies the determinant formula's first part.

Step 5: Simplify the Second Term

Calculate \( -z((z) + xy) = -z(z + xy) = -z^2 - xyz \).

Step 6: Simplify the Third Term

Calculate \( -y((-z)(-x) - y) = -y(zx - y) = -y(zx) + y^2 = -xyz + y^2 \).

Step 7: Combine and Simplify Terms

Combine all simplified terms to find the determinant: \( (1 + x^2) + (-z^2 - xyz) + (-xyz + y^2) \)Which simplifies to:\( 1 + x^2 - z^2 - 2xyz + y^2 \).

Step 8: Use Given Equation

Since \( x^2 + y^2 + z^2 = 1 \), substitute it into the expression:\( 1 + (x^2 + y^2 + z^2 - z^2) - 2xyz = 1 + (1 - z^2) - 1 - 2xyz = 1 - z^2 + x^2 + y^2 - 2xyz \).

Step 9: Final Simplification

Notice that \( x^2 + y^2 + z^2 = 1 \) makes \( 1 - z^2 + x^2 + y^2 = 1 \), hence the expression simplifies to \( 1 - z^2 + 1 - 2xyz = 2 - 2xyz \).

Step 10: Final Result

Thus, the value of the determinant is \( 2 - 2xyz \), which corresponds to option (d).

Key Concepts

Matrix Algebra

Matrix algebra forms the foundation of many mathematical concepts, including determinants, which are used for multiple applications in fields like engineering and computer science. A matrix is essentially a rectangular array of numbers, set out in rows and columns. These numbers are called elements or entries. The process of matrix algebra involves operations such as addition, subtraction, and multiplication of matrices. These operations allow us to manipulate and analyze complex datasets in a structured manner.

Matrices are particularly useful when solving linear equations or transforming geometric data. Understanding matrix algebra is crucial for tackling problems involving multivariable systems or dealing with data transformations in machine learning models. In the context of this problem, matrix algebra helps in determining the orientation and volume scaling factor by calculating determinants. The determinant provides essential information, such as whether a matrix is invertible or not, aiding further algebraic manipulations.

3x3 Determinant Formula

The 3x3 determinant formula is a pivotal tool in matrix algebra. For a 3x3 matrix \( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant \( |A| \) is calculated using the formula: \[ a(ei - fh) - b(di - fg) + c(dh - eg), \] where each term expands based on specific elements from the matrix.

This formula is crucial because it allows you to calculate the determinant, a value that provides significant insights. For instance, the determinant of a 3x3 matrix can determine if a system of equations has a unique solution.

• If the determinant is zero, the matrix is singular, indicating that the system of equations either has no solution or infinitely many solutions.
• If the determinant is non-zero, the matrix is non-singular, meaning the system of equations has a unique solution.

Understanding and using the determinant formula empowers students to tackle problems involving matrices with confidence and precision.

Matrix Elements Identification

Identifying the elements in a matrix is a simple yet crucial step when calculating determinants or performing matrix operations. Every matrix is made up of individual elements, each with its specific position indicated by its row and column.

Let’s take our given matrix: \[ \begin{pmatrix} 1 & z & -y \-z & 1 & x \y & -x & 1 \end{pmatrix}, \] where each element needs to be recognized for substitution into formulas. These elements are: - \( a = 1 \), \( b = z \), \( c = -y \) - \( d = -z \), \( e = 1 \), \( f = x \) - \( g = y \), \( h = -x \), \( i = 1 \)

By correctly identifying each element, one can accurately apply determinant formulas or perform the necessary algebraic operations. This provides clarity and reduces errors in calculations. Once you've identified each element, the path to solving the determinant becomes clearer and more manageable.