Problem 49 - Algebra and Trigonometry

Question

\(47-50=\) 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 solution is \(x = 1\).

Step 1: Understand the Problem

We need to find the value of \(x\) that makes the determinant of the given matrix equal to zero.

Step 2: What is a Determinant?

For a 3x3 matrix, the determinant can be found using a formula which involves multiplying and subtracting products of the elements of the matrix.

Step 3: Compute the Determinant

Compute the determinant of the matrix \( \begin{vmatrix} 1 & 0 & x \ x^2 & 1 & 0 \ x & 0 & 1 \end{vmatrix} = 1(1\cdot1 - 0\cdot0) - 0(x^2\cdot1 - x\cdot0) + x(x^2\cdot0 - x\cdot1) \).

Step 4: Simplify the Determinant

The expression simplifies to \(1 - x^3\). Therefore, the determinant is \(1 - x^3 = 0\).

Step 5: Solve for x

Set the equation \(1 - x^3 = 0\) equal to zero and solve for \(x\): \( x^3 = 1 \).

Step 6: Find x

The solution to \(x^3 = 1\) is \(x = 1\).

Key Concepts

Understanding a 3x3 Matrix

A 3x3 matrix is a type of square matrix that includes three rows and three columns. Each element is arranged in rows and columns making it part of a larger matrix structure. In mathematical terms, a 3x3 matrix can be denoted as a grid of elements as follows:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]Each of the letters \(a\) through \(i\) represents an element, or an entry, of the matrix. These elements are crucial, as they will be used to compute the determinant of the matrix. In practice:

• The positions of these elements in the matrix are very important.
• Typically, operations on matrices, like finding determinants or inverses, depend on these positions.
• 3x3 matrices are often used in real-world applications like solving systems of equations or transformations in graphics.

If you picture each position filling in, you can imagine how these are manipulated according to matrix rules to extract useful mathematical information.

Solving Equations Using Determinants

To solve for unknowns in equations using determinants, you rely on methods from linear algebra. Specifically, when finding solutions for variables like \(x\), you frequently need the determinant.The concept behind solving equations with determinants starts with setting up the determinant equal to zero. For the problem at hand:

• The equation comes from setting the determinant of the matrix to zero, as it needs to be evaluated for specific values of \(x\).
• The process involves simplifying the determinant expression.
• After simplification, mathematically solving for \(x\) follows conventional algebraic rules.

This is an essential concept in linear algebra, as it helps find solutions for systems of equations by identifying when matrices do not have an inverse (i.e., when their determinant is zero).

Fundamentals of Matrix Algebra

Matrix algebra involves diverse operations such as addition, subtraction, and multiplication, along with special procedures like finding inverses and computing determinants:

• Addition and Subtraction: Two matrices of the same size can be added or subtracted by adding or subtracting corresponding elements.
• Multiplication: Matrix multiplication is more complex. The number of columns in the first matrix must match the number of rows in the second.
• Determinants: The determinant is a special number associated with square matrices that can provide insights into the properties of the matrix, like whether it’s invertible.
• Inverse: Another matrix operation is finding the inverse, often used to solve systems of linear equations, but only possible if the determinant is not zero.

In summary, matrix algebra serves as the backbone for many linear algebra problems. Understanding these basics allows us to tackle more complex mathematical problems, including those involving the determinants of 3x3 matrices.