Problem 31 - A Graphical Approach to College Algebra

Question

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rrr}4 & 3 & 0 \\\2 & 0 & 1 \\\\-3 & x & -1\end{array}\right]=5$$

Step-by-Step Solution

Answer

\( x = -2 \)

Step 1: Understand the determinant formula for a 3x3 matrix

The determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is evaluated using the formula: \( a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 2: Identify the elements of the matrix

The given matrix is \( \begin{bmatrix} 4 & 3 & 0 \ 2 & 0 & 1 \ -3 & x & -1 \end{bmatrix} \). The elements are \( a = 4 \), \( b = 3 \), \( c = 0 \), \( d = 2 \), \( e = 0 \), \( f = 1 \), \( g = -3 \), \( h = x \), and \( i = -1 \).

Step 3: Calculate intermediate terms for the determinant expression

Calculate the products: \( ei - fh = 0(-1) - 1(x) = -x \), \( di - fg = 2(-1) - 1(-3) = -2 + 3 = 1 \), and \( dh - eg = 2x - 0(-3) = 2x \).

Step 4: Substitute into the determinant formula

Using the determinant formula: \( a(ei - fh) - b(di - fg) + c(dh - eg) \), substitute the values: \( 4(-x) - 3(1) + 0(2x) = -4x - 3 \).

Step 5: Set the determinant equal to 5

The problem states that the determinant of the matrix is 5. Therefore, we set the expression \(-4x - 3\) equal to 5: \(-4x - 3 = 5\).

Step 6: Solve for \(x\)

Add 3 to both sides to isolate the term with \(x\): \(-4x = 8\). Then divide both sides by -4 to find \(x\): \(x = -2\).

Key Concepts

3x3 matrix

A 3x3 matrix is a grid consisting of three rows and three columns, filled with numerical values. It is a fundamental structure in linear algebra, widely used in various calculations, including solving systems of equations and transforming geometric data. Each spot in this 3x3 array is called an element.
The specific arrangement of entries can greatly impact calculations such as determinants and inverses. 3x3 matrices are more manageable chunks of data, compared to larger matrices, making them ideal for educational contexts.
In the given exercise, the matrix is:

• Row 1: [4, 3, 0]
• Row 2: [2, 0, 1]
• Row 3: [-3, x, -1]

matrix elements

Matrix elements are the individual numbers that make up a matrix. Each element serves a critical role when performing operations such as finding a determinant. In a 3x3 matrix, there are nine elements, identified by their row and column positions.
For example, in the given exercise:

• Element a = 4 (Row 1, Column 1)
• Element b = 3 (Row 1, Column 2)
• Element c = 0 (Row 1, Column 3)
• Element d = 2 (Row 2, Column 1)
• Element e = 0 (Row 2, Column 2)
• Element f = 1 (Row 2, Column 3)
• Element g = -3 (Row 3, Column 1)
• Element h = x (Row 3, Column 2)
• Element i = -1 (Row 3, Column 3)

Understanding matrix elements is essential, as they are directly used in applying determinant formulas and other matrix operations.

determinant formula

The determinant is a special number assigned to a square matrix, revealing key properties like invertibility. For a 3x3 matrix, the determinant is calculated using a specific formula:
\[ a(ei - fh) - b(di - fg) + c(dh - eg) \]
Here, each term represents a minor of the matrix, expanded using the first row. This formula illustrates how various submatrices contribute to the total determinant.
In the exercise, substituting the values yields:

• \( 4(0(-1) - 1(x)) - 3(2(-1) - 1(-3)) + 0(2x - 0(-3)) \)
• This simplifies to: \(-4x - 3\)

The determinant being equal to 5 sets up an equation to solve for \(x\).

linear algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between them. It includes topics like matrices, determinants, vectors, and systems of linear equations. This field forms the backbone of many scientific disciplines, providing tools to model real-world problems.
The determinant is a crucial concept in linear algebra, used to determine whether a matrix has an inverse, as well as in calculating eigenvalues and solving systems of equations. Understanding how to manipulate and calculate determinants enriches one's comprehension of linear transformations and other advanced linear algebra concepts.
This exercise demonstrates the process of solving a matrix determinant equation, showcasing the practical applications of linear algebra in mathematical problem-solving.