Problem 63 - Precalculus: Mathematics for Calculus

Question

Solve for \(x\) $$\left|\begin{array}{ccc} x & 12 & 13 \\ 0 & x-1 & 23 \\ 0 & 0 & x-2 \end{array}\right|=0$$

Step-by-Step Solution

Answer

The solutions for \( x \) are 0, 1, and 2.

Step 1: Identify the Matrix

The matrix given in the problem is an upper triangular matrix, which is a square matrix where all the elements below the diagonal are zero. In this case, the matrix is a 3x3 upper triangular matrix.

Step 2: Recall the Determinant Property of Upper Triangular Matrices

For an upper triangular matrix, the determinant is the product of its diagonal elements. Therefore, the determinant of the given matrix is \( x \cdot (x-1) \cdot (x-2) \).

Step 3: Set the Determinant to Zero

We are asked to solve for \( x \) such that the determinant of the matrix equals zero. Therefore, we set the product of the diagonal elements to zero: \( x(x-1)(x-2) = 0 \).

Step 4: Solve the Equation

Set each factor of the product equal to zero: \( x = 0 \), \( x - 1 = 0 \), or \( x - 2 = 0 \). Solving these gives \( x = 0 \), \( x = 1 \), and \( x = 2 \).

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special kind of square matrix. In this type of matrix, all the elements below the main diagonal are zero. The main diagonal is the set of elements that stretch from the top-left corner to the bottom-right corner of the matrix. Imagine a staircase pattern where everything below it disappears! Simple, right? Here are some features of upper triangular matrices:

• All elements directly below the diagonal are always zero, which makes calculations easier.
• They often arise in problems involving linear equations and are very useful for solving systems of equations.
• You can find them naturally appearing during certain matrix transformations, like Gaussian elimination.

This specific arrangement simplifies many matrix operations, including calculating the determinant. Knowing these properties helps tackle many mathematics problems efficiently.

Solving Equations

When we talk about solving equations in the context of matrices, we're often referring to finding values for variables that make an equation true. In the case of our upper triangular matrix, we worked with its determinant. Here's how we use it:

• The determinant of a matrix provides important properties, like telling us if the matrix is invertible.
• For upper triangular matrices, the determinant is simply the product of the diagonal elements: no need for complex calculations!
• To solve the equation, we set this determinant equal to zero and find the values of the variable involved.

In our exercise, this process involved setting our product to zero and solving it like any standard equation. This resulted in straightforward solutions for the variable \(x\). Breaking it down into factors alternating through zero was key, which helped us identify when this matrix becomes singular (a key concept!).

Matrices

Matrices are fascinating mathematical objects used to organize numbers in rows and columns. They are a powerful tool in everything from simple math to complex computations. Here's why matrices are so important:

• They represent systems of linear equations, which can be used to solve complex problems in physics, computer science, and many other fields.
• Matrices can be manipulated through operations like addition, multiplication, and finding determinants, which help solve these systems.
• Special types of matrices, such as upper triangular matrices, can simplify calculations. Their structure reduces the complexity of finding solutions.

In our exercise, we were given a 3x3 matrix. By leveraging its upper triangular form and calculating its determinant, we were able to solve for \(x\). Understanding how these arrays of numbers interact provides deep insights into both theoretical and applied mathematics. It's like building blocks for equations, and they can form the basis for solving many real-world problems!