Problem 13 - Algebra for College Students

Question

For Problems 13-28, evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}1 & 2 & -1 \\ 3 & 1 & 2 \\ 2 & 4 & 3\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant of the matrix is -25.

Step 1: Identify the Matrix

Let's denote the given matrix as \(A\). The matrix \(A\) is:\[A = \begin{bmatrix} 1 & 2 & -1 \ 3 & 1 & 2 \ 2 & 4 & 3 \end{bmatrix}\] The task is to find the determinant of this \(3 \times 3\) matrix.

Step 2: Understand the Determinants Formula

The formula for the determinant of a \(3 \times 3\) matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\) is given by:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\] We'll apply this formula to our matrix.

Step 3: Assign Variables to Elements of the Matrix

Assign elements of the matrix \(A\) to the variables, where:\[ a = 1, b = 2, c = -1 \ d = 3, e = 1, f = 2 \ g = 2, h = 4, i = 3 \]

Step 4: Apply the Determinant Formula

Plug the values from the matrix into the determinant formula:\[\text{det}(A) = 1(1 \times 3 - 2 \times 4) - 2(3 \times 3 - 2 \times 2) + (-1)(3 \times 4 - 1 \times 2)\]Simplify each expression.

Step 5: Simplify the Determinant Terms

Calculate each part of the expression:1. \(1(3 - 8) = 1 \times (-5) = -5\)2. \(-2(9 - 4) = -2 \times 5 = -10\)3. \(-1(12 - 2) = -1 \times 10 = -10\)

Step 6: Calculate the Determinant Value

Add up the results of the calculations:\[\text{det}(A) = -5 - 10 - 10 = -25\] The determinant of the matrix is \(-25\).

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental aspect of linear algebra that deals with computations involving matrices. Matrices are arrays of numbers organized into rows and columns, which can represent various types of data, transformations, or systems of equations. Understanding how to work with matrices allows for operations such as addition, subtraction, and multiplication, as well as finding the inverse and the determinant of a matrix.

One crucial operation in matrix algebra is finding the determinant. The determinant is a scalar value that can provide important insights about a matrix, including whether it's invertible or singular (non-invertible). For a matrix to have an inverse, its determinant must be non-zero.

When tackling problems involving determinants, remember:

• Matrix addition and subtraction involve adding or subtracting corresponding elements.
• Matrix multiplication isn't element-wise; instead, it combines rows and columns.
• Determinants require a specific formula, particularly as the matrix size increases.

Mastering these operations will give students the flexibility to solve complex algebraic problems and understand a wide range of mathematical applications.

3x3 Matrix

A 3x3 matrix is a specific type of square matrix with three rows and three columns. The clarity of its size means it contains exactly nine elements. These matrices are particularly common when dealing with systems of linear equations that have three variables or when representing transformations in three-dimensional space.

In the exercise provided, the matrix:
\[A = \begin{bmatrix} 1 & 2 & -1 \ 3 & 1 & 2 \ 2 & 4 & 3 \end{bmatrix} \]
is a perfect example of a 3x3 matrix. Each element in such a matrix can be referenced by its position, using a two-number system representing its row and column.

For example:

• The element at row 1, column 1 is "1".
• The element at row 2, column 3 is "2".

The calculation of its determinant involves leveraging these positions along with the determinant formula specific to 3x3 matrices. Understanding properly how these positions interact is key to ensuring accurate calculations. Practicing with 3x3 matrices can significantly enhance a student's confidence in higher-level mathematics.

Properties of Determinants

The properties of determinants provide powerful tools when working with matrices, especially when simplifying calculations or understanding the characteristics of matrix transformations.

For a 3x3 matrix, like the one in our exercise, the determinant can be interpreted as the scaling factor of the linear transformation described by the matrix. If the determinant is zero, it means the transformation collapses into a lower dimension, indicating that the matrix is singular and has no inverse.

Key properties include:

• The determinant of the identity matrix of any size is 1.
• Swapping two rows or columns of a matrix results in the determinant changing sign.
• If a matrix row or column is multiplied by a scalar, the determinant is scaled by that same factor.
• A matrix is singular if its determinant is zero.

These properties assist in understanding how various changes to a matrix affect its determinant, making them invaluable for checking the correctness of complex calculations and exploring deeper applications in linear algebra.