Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rrr}2 x & 1 & -1 \\\0 & 4 & x \\\3 & 0 & 2\end{array}\right]=x$$

Step-by-Step Solution

Answer

The value of \(x\) is 0.

Step 1: Understand the Determinant of a 3x3 Matrix

The determinant of a 3x3 matrix is calculated using the rule of Sarrus or the cofactor expansion. This involves taking the sum of the products of diagonals and subtracting the sum of the products of the anti-diagonals.

Step 2: Set Up the Determinant Equation

The given matrix is:\[\begin{bmatrix}2x & 1 & -1 \0 & 4 & x \3 & 0 & 2\end{bmatrix}\]We need to calculate the determinant of this matrix and set it equal to \(x\):\[\operatorname{det}(A) = x\]

Step 3: Calculate the Determinant

Using the formula for the determinant of a 3x3 matrix, calculate step by step:\[\operatorname{det}(A) = 2x(4 \cdot 2) - 1(0 \cdot 2) + (-1)(0 \cdot 3) + x(1 \cdot 3) + 0 - 3(1 \cdot x)\]Simplify the terms:\[= 2x \cdot 8 + 0 + 0 + 3x - 3x\]Calculate to get:\[16x\]

Step 4: Solve the Equation for x

Set the determinant equal to \(x\):\[16x = x\]Subtract \(x\) from both sides to isolate \(x\):\[15x = 0\]Solve for \(x\):\[x = 0\]

Step 5: Verify the Solution

After obtaining \(x = 0\), substitute back into the original equation to ensure there are no contradictory results. The original equation simplifies correctly, confirming the solution.

Key Concepts

3x3 matrix

A 3x3 matrix is a square array consisting of three rows and three columns. It's commonly used in various mathematical and scientific applications.
This type of matrix can be represented as:

• The first row: \([a_{11}, a_{12}, a_{13}]\)
• The second row: \([a_{21}, a_{22}, a_{23}]\)
• The third row: \([a_{31}, a_{32}, a_{33}]\)

In our context, the 3x3 matrix is used to solve determinant equations that are essential in linear algebra. The determinant is a special number that helps determine the properties of the matrix and solve systems of linear equations. Understanding how to handle a 3x3 matrix is crucial for more advanced topics in linear algebra.
For proficiency in handling these matrices, start by mastering basic matrix operations such as addition and multiplication.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method for finding the determinant of a matrix. In a 3x3 matrix, it involves evaluating each minor formed by deleting a row and a column, then adjusting it by a checkerboard pattern of signs (alternating positive and negative).
For the given matrix: \[\begin{bmatrix}2x & 1 & -1 \0 & 4 & x \3 & 0 & 2\end{bmatrix}\] The determinant can be computed by expanding along any row or column.
Let's choose the first row for simplicity:

• Multiply the element by the determinant of the minor matrix it forms (2x times the minor of element 4,2)
• Repeat for all elements in the row

The signs alternate starting with positive for the first element of the chosen row or column. Understanding this process is crucial because it contributes to ways we solve systems of linear equations.

Determinant Calculation

Determinant calculation of a 3x3 matrix can be done through cofactor expansion or by using the rule of Sarrus, a direct method involving diagonals.For the provided example, cofactor expansion involves elaborately simplifying the expression:

• Compute the product of elements in diagonals and anti-diagonals.
• Subtract the anti-diagonal products from the diagonal products.

The specific calculation detailed in the steps provided helped us arrive at the equation: \[16x = x\]
This simplification deduced the final value to solve for \(x\).While challenging at first, practice with different approaches will help improve your efficiency and understanding of determinant calculation.

Linear Algebra

Linear Algebra is the branch of mathematics that deals with vectors, vector spaces, linear transformations, and matrices. It provides powerful techniques for solving systems of linear equations. A major part of this involves matrices and determinants. Understanding determinants plays a vital role in Linear Algebra because:

• They help in finding out if a matrix is invertible.
• They provide information about eigenvalues and the linear transformations represented by matrices.

By calculating the determinant and interpreting its value, we determine important properties about the matrix and any equations it might be part of.
Linear Algebra is foundational to various fields such as engineering, physics, computer science, and more. Familiarity with determinants, particularly in 3x3 matrices, is an essential skill for students and professionals in these areas.