Problem 30 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Understand the Problem

We need to find the value of \(x\) such that the determinant of the given 3x3 matrix equals 3.

Step 2: Write the Determinant Formula

For a 3x3 matrix \(\begin{pmatrix}a&b&c \ d&e&f \ g&h&i \end{pmatrix}\), the determinant is calculated by \(a(ei-fh) - b(di-fg) + c(dh-eg)\).

Step 3: Identify Matrix Elements

Identify the elements of our matrix:\[ a = -2, \ b = 0, \ c = 1, \ d = -1, \ e = 3, \ f = x, \ g = 5, \ h = -2, \ i = 0. \]

Step 4: Calculate Each Component

Substitute these values into the determinant formula:- \(ei - fh = 3 \cdot 0 - x \cdot (-2) = 2x\)- \(di - fg = (-1) \cdot 0 - x \cdot 5 = -5x\)- \(dh - eg = (-1)(-2) - 3 \cdot 5 = 2 - 15 = -13\)

Step 5: Substitute and Solve

Substitute into the formula:\[-2(2x) - 0(-5x) + 1(-13) = 3.\]Simplify to get:\[-4x - 13 = 3.\]

Step 6: Solve for x

Rearrange the equation to solve for \(x\):\[-4x - 13 = 3 \]\[-4x = 3 + 13 \]\[-4x = 16 \]\[x = -4.\]

Step 7: Verify the Solution

Verify the calculation by substituting \(x = -4\) back into the determinant equation and checking if it equals 3: - Substitute \(-4\) into each step calculation and ensure the operations are correct.- The determinant should equal 3.

Key Concepts

3x3 Matrix

A 3x3 matrix is a rectangular array consisting of three rows and three columns. Each entry in the matrix is known as an element, and the matrix itself can be represented in the form:\[\begin{pmatrix}a & b & c \d & e & f \g & h & i\\end{pmatrix}\]This structure is fundamental in linear algebra and is used to solve various mathematical problems involving systems of equations, transformations, and more.

• Rows: The horizontal lines of elements. In a 3x3 matrix, there are three rows.
• Columns: The vertical lines of elements, also three in number in a 3x3 matrix.
• Elements: Each number or variable within the matrix.

The problem we are addressing includes such a matrix where the elements contribute to calculating the determinant.

Determinant Formula

The determinant of a 3x3 matrix is a special scalar value that can provide insights into the matrix, such as whether it is invertible. To find the determinant of a 3x3 matrix, use the following formula:\[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula utilizes the elements of the matrix from their respective positions marked as follows:

• \(a, b, c\)
• \(d, e, f\)
• \(g, h, i\)

This mathematical operation involves calculating three different minor determinants from the matrix, then combining these calculations by factoring in element positions. It gives us a single value representative of the matrix's properties.

Matrix Elements

Matrix elements are the individual numbers or variables that make up the matrix. In our specific example, the elements of the matrix are identified as:

• \(a = -2\)
• \(b = 0\)
• \(c = 1\)
• \(d = -1\)
• \(e = 3\)
• \(f = x\)
• \(g = 5\)
• \(h = -2\)
• \(i = 0\)

Each element serves a functional role within the determinant formula, directly impacting the resulting determinant value. Understanding their position and value is crucial in performing calculations or solving equations.

Algebraic Equations

Algebraic equations involve finding unknown variables through the manipulation of constants and variables within expressions. In our exercise, the strategy is to solve for the unknown variable \(x\).
The determinant equation is initially given as:\[-2(2x) - 13 = 3\]This equation represents the process of substituting identified matrix elements into the determinant formula, resulting in an expression with \(x\). Solving an algebraic equation generally includes steps such as:

• Isolating the variable
• Performing arithmetic operations to simplify
• Rewriting the equation to reach the desired result

Ultimately, this method led us to calculate and verify that \(x = -4\), ensuring the determinant equals the given value of 3.