Problem 65 - Precalculus: Mathematics for Calculus

Question

Solve for \(x\) $$\left|\begin{array}{lll} 1 & 0 & x \\ x^{2} & 1 & 0 \\ x & 0 & 1 \end{array}\right|=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 1 \) and \( x = -1 \).

Step 1: Identify the Task

We need to find the value of \( x \) for which the determinant of a 3x3 matrix is equal to zero.

Step 2: Write the Determinant Formula

The matrix is \(\begin{pmatrix}1 & 0 & x \x^2 & 1 & 0 \x & 0 & 1\end{pmatrix}\). We will use the formula for the determinant of a 3x3 matrix: \(\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\).

Step 3: Find the Determinant

Using the formula and based on the matrix:1. \( a = 1 \), \( b = 0 \), \( c = x \)2. \( d = x^2 \), \( e = 1 \), \( f = 0 \)3. \( g = x \), \( h = 0 \), \( i = 1 \)Calculate each term:- \( ei-fh = 1\times 1 - 0\times 0 = 1 \)- \( di-fg = x^2\times 1 - 0\times x = x^2 \)- \( dh-eg = x^2\times 0 - 1\times x = -x \)Substitute these into the determinant formula to get:\( 1(1) - 0(x^2) + x(-x) \).

Step 4: Simplify the Expression

This simplifies to \( 1 - x^2 \). Set this equal to zero since the determinant is given as zero:\[ 1 - x^2 = 0 \]

Step 5: Solve the Equation for x

Move \( x^2 \) to the other side of the equation:\[ x^2 = 1 \]Take the square root of both sides:\[ x = \pm 1 \]

Step 6: Final Solution

The solutions to the equation are \( x = 1 \) and \( x = -1 \).

Key Concepts

Understanding the 3x3 Matrix

A 3x3 matrix is a square array containing nine elements arranged in three rows and three columns. These matrices are common in linear algebra and play a critical role in various mathematical computations. You might encounter them when solving systems of equations, finding eigenvalues, or in transformations in space.
To represent a 3x3 matrix, we employ the following convention:

• Let the matrix elements be denoted using lowercase letters like a, b, c,…, i.
• These elements fill the matrix row by row from top to bottom.

Here's an example structure of a 3x3 matrix:\[\begin{pmatrix}a & b & c \d & e & f \g & h & i\end{pmatrix}\]
You can compute the determinant of any 3x3 matrix using a specific formula, which helps in solving problems like finding solutions to systems of equations. The determinant provides important information on whether a matrix is invertible. For our exercise, understanding the structure of the matrix helps set the stage for solving the equation.

The Process to Solve for x

Solving for \( x \) in a matrix involves setting its determinant equal to zero or another specified value. In the given exercise, we encountered a task where the matrix's determinant needed to equal zero.
Here's how we went about solving for \( x \):

• We used the determinant formula for a 3x3 matrix, which involves computing specific products and sums of the matrix's elements.
• The result is an equation that can be resolved to find the value of \( x \).

For this problem, we identified the equation:\[ 1 - x^2 = 0 \]Then, the task was to solve for \( x \):

• Rearrange terms to isolate \( x^2 \), yielding \( x^2 = 1 \).
• Find the square root of both sides to solve for \( x \), resulting in two solutions: \( x = 1 \) and \( x = -1 \).

Each step in this process helps narrow down the potential values for \( x \) that solve the equation derived from the determinant formula.

Exploring Matrix Algebra

Matrix algebra is a fundamental component of linear algebra, dealing with operations involving matrices, such as addition, multiplication, and finding determinants. These operations make it possible to solve complex mathematical problems efficiently.
When understanding how to apply matrix algebra for determinants, remember:

• Determinants give you a scalar value that can help determine if a matrix is invertible.
• The formula for a 3x3 determinant involves selecting a row or column, computing the minor determinants, and combining them with assigned signs.

Here’s the general form for computing a 3x3 determinant:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Exploring matrix algebra requires familiarity with these operations and the inter-relationship of matrix components. Adeptness in using this algebraic toolkit allows you to deal with numerous mathematical models across diverse disciplines like physics and engineering, thereby making it crucial for your educational toolkit.