Problem 48 - Algebra and Trigonometry

Question

\(47-50=\) Solve for \(x\) $$ \left|\begin{array}{lll}{x} & {1} & {1} \\ {1} & {1} & {x} \\ {x} & {1} & {x}\end{array}\right|=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Set Up the Determinant Equation

We start by writing the equation for the determinant of a 3x3 matrix. The given matrix is: \[ \begin{bmatrix} x & 1 & 1 \ 1 & 1 & x \ x & 1 & x \end{bmatrix} \] We need to find the determinant of this matrix and set it equal to zero.

Step 2: Compute the Determinant

To find the determinant of the 3x3 matrix, use the formula: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] Substituting the elements from our matrix: \[ x(1 \times x - 1 \times 1) - 1(x \times x - 1 \times x) + 1(x \times 1 - 1 \times 1) \] This simplifies to: \[ x(x - 1) - (x^2 - x) + (x - 1) \] Simplify further: \[ x^2 - x - x^2 + x + x - 1 \] Finally, we have: \[ -1 + x \]

Step 3: Solve the Determinant Equation

Set the simplified determinant equal to zero: \[ -1 + x = 0 \] Solve for \(x\): \[ x = 1 \]

Key Concepts

Understanding a 3x3 Matrix

A 3x3 matrix is a rectangular array consisting of three rows and three columns. In a matrix, elements are ordered pairs, meaning each entry is positioned in a specific row and column, like a map grid. Matrices help us represent and solve complex linear equations.

• In our exercise, the matrix is written as:\[\begin{bmatrix} x & 1 & 1 \ 1 & 1 & x \ x & 1 & x \end{bmatrix}\]
• The numbers in this matrix, including \(x\), can represent coefficients in a system of equations. Here, we're solving for the unknown \(x\).
• The location of each number is crucial, as it affects how we compute the determinant and later solve the equation.

The 3x3 matrix is a foundational block in linear algebra, useful in multiple fields including physics, engineering, and computer science. Understanding the arrangement and significance of elements within a matrix is essential when delving further into concepts like determinants and matrix operations.

Solve for X in a Determinant Equation

Solving for \(x\) in matrix equations often involves finding the determinant, especially to check singular matrices where the determinant equals zero.

• The determinant of a matrix is a special number that we calculate from its elements. For a 3x3 matrix, calculating this involves a specific setback formula.
• Our exercise gives us a matrix and requires setting its determinant to zero since the system can have a unique solution only if the matrix is not invertible.
• Computation starts by identifying and substituting elements into the determinant formula: \( a(ei - fh) - b(di - fg) + c(dh - eg) \). We insert values from the matrix and simplify.
• The critical step is simplifying the expression, leading us to an equation: \(-1 + x = 0\), which lets us solve for \(x\).

Performing this operation accurately helps us find right answers in real-world problems. Remember, equating the determinant to zero implies a system of equations has either no solutions or infinite solutions.

Basics of Matrix Algebra

Matrix algebra involves operations similar to regular algebra but applied to entire matrices. Determinants are a key aspect in matrix algebra, revealing properties of the linear transformations they represent.

• Understanding matrix algebra is essential in science and engineering. It involves addition, subtraction, and multiplication of matrices along with finding inverses and determinants.
• Our core focus here is on determining the determinant and solving simple matrix equations like finding \(x\). While determinant calculations can seem complex, breaking them into smaller steps like element substitution and simplification makes them manageable.
• Determinants help us determine the invertibility of a matrix or understand its geometric properties, like scaling or rotating data in different applications.
• This type of algebra isn't restricted to theoretical mathematics; it extends to practical applications in computer graphics, data science, and systems analysis where multi-dimensional datasets are common.

Familiarity with matrix operations and the role of determinants is a significant step into deeper mathematical and practical applications.