Problem 14 - Elementary and Intermediate Algebra

Question

Evaluate. $$ \left|\begin{array}{rrr} {2} & {-1} & {1} \\ {1} & {2} & {-1} \\ {3} & {4} & {-3} \end{array}\right| $$

Step-by-Step Solution

Answer

-6

Step 1: Identify the Matrix and its Elements

Identify the matrix elements. Here, the given matrix is: \[ \begin{pmatrix} 2 & -1 & 1 \ 1 & 2 & -1 \ 3 & 4 & -3 \end{pmatrix} \]

Step 2: Use the Formula for Determinant of a 3x3 Matrix

To find the determinant of a 3x3 matrix, use the rule: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where \[ A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \]

Step 3: Substitute the Corresponding Values

Substitute the values from the matrix into the formula. Here: \[ a = 2, b = -1, c = 1, d = 1, e = 2, f = -1, g = 3, h = 4, i = -3 \]

Step 4: Calculate Each Term

Calculate each part of the formula: \[ (ei - fh) = (2 \times -3 - (-1) \times 4) = -6 + 4 = -2 \] \[ (di - fg) = (1 \times -3 - (-1) \times 3) = -3 + 3 = 0 \] \[ (dh - eg) = (1 \times 4 - 2 \times 3) = 4 - 6 = -2 \]

Step 5: Combine the Results

Now, combine the calculated parts into the determinant formula: \[ \text{det}(A) = 2(-2) - (-1)(0) + 1(-2) = -4 + 0 - 2 = -6 \]

Key Concepts

Matrix Evaluation

Understanding matrix evaluation is important in linear algebra. A matrix is simply a rectangular array of numbers arranged in rows and columns. In many mathematical problems, we need to evaluate a matrix, which involves performing specific operations on its elements.

When we discuss evaluation, we often refer to finding the determinant. The determinant offers us valuable information about the matrix, such as whether it is invertible or the area of a parallelogram defined by the matrix's vectors if considering a 2x2 matrix. For the 3x3 matrix in our problem, the determinant helps us understand several properties, including the system of equations the matrix might represent.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. Matrices are fundamental objects in linear algebra.

They allow us to represent and manipulate linear equations efficiently. In our example, we're dealing with a 3x3 matrix, a common type used in various applications from computer graphics to solving systems of equations.

By evaluating matrices and using linear algebra techniques, we can simplify complex problems to understand relationships between different linear systems. The determinant we calculated for the 3x3 matrix in the problem is a key concept in linear algebra that helps us determine the matrix's properties.

Determinant Formula

The determinant of a matrix is a special number that can reveal insights about the matrix. For a 3x3 matrix, the formula for finding the determinant might seem complicated at first, but it's manageable once understood.

The determinant for a 3x3 matrix is given by:

\[\begin{equation} \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \text{ where } A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \end{equation}\]

This formula involves breaking down the matrix into smaller 2x2 determinants and combining them with respect to their positions in the initial matrix.

For our specific matrix: \begin{pmatrix} 2 & -1 & 1 \ 1 & 2 & -1 \ 3 & 4 & -3 \end{pmatrix},the variables are:

• a = 2
• b = -1
• c = 1
• d = 1
• e = 2
• f = -1
• g = 3
• h = 4
• i = -3

Substituting these values into the formula helps us calculate the determinant step-by-step.

Step-by-Step Solution

We broke down the determinant calculation into manageable steps.
Here's an easy way to understand the solution. Let's follow the steps together.

First, identify the matrix elements from:\begin{pmatrix}2 & -1 & 1 \ 1 & 2 & -1 \ 3 & 4 & -3 \end{pmatrix}.Now, apply the determinant formula for a 3x3 matrix: \[\begin{equation} \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \end{equation}\].Next, substitute the values into the formula:1. Calculate (ei - fh): (2 \times -3 - (-1) \times 4) = -6 + 4 = -2.2. Compute (di - fg): (1 \times -3 - (-1) \times 3) = -3 + 3 = 0.3. Determine (dh - eg): (1 \times 4 - 2 \times 3) = 4 - 6 = -2.Finally, substitute these results back into the determinant formula: \[\begin{equation}\text{det}(A) = 2(-2) - (-1)(0) + 1(-2) = -4 + 0 - 2 = -6. \end{equation}\]So the determinant of the given 3x3 matrix is -6.