Problem 14 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{rrr} 2 & -5 & 1 \\ -3 & 1 & 6 \\ 4 & -2 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is -133.

Step 1: Identify the Matrix

The matrix given is a 3x3 square matrix:\[\begin{bmatrix} 2 & -5 & 1 \ -3 & 1 & 6 \ 4 & -2 & 3\end{bmatrix}\]

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

The determinant of a 3x3 matrix \(A\) with elements \(a_{ij}\) can be calculated using the formula:\[\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

Step 3: Substitute Values from the Matrix into the Formula

Substitute the values: \(a_{11} = 2\), \(a_{12} = -5\), \(a_{13} = 1\), \(a_{21} = -3\), \(a_{22} = 1\), \(a_{23} = 6\), \(a_{31} = 4\), \(a_{32} = -2\), \(a_{33} = 3\).\[\begin{aligned}\text{det}(A) = & \ 2(1 \times 3 - 6 \times (-2)) \ & -(-5)((-3) \times 3 - 6 \times 4) \ & + 1((-3) \times (-2) - 1 \times 4)\end{aligned}\]

Step 4: Calculate Each Term

Calculate each term separately:1. \(2(1 \times 3 - 6 \times (-2)) = 2(3 + 12) = 2 \times 15 = 30\)2. \(-(-5)((-3) \times 3 - 6 \times 4) = 5(-9 - 24) = 5 \times (-33) = -165\)3. \(1((-3) \times (-2) - 1 \times 4) = 1(6 - 4) = 1 \times 2 = 2\)

Step 5: Sum the Calculated Values

Add the results from Step 4 together to obtain the determinant:\[\text{det}(A) = 30 + (-165) + 2 = 30 - 165 + 2 = -133\]

Key Concepts

3x3 Matrix

A 3x3 matrix is a rectangular array consisting of three rows and three columns of numbers. In linear algebra, matrices are fundamental objects used to represent systems of linear equations, transformations, or any relation characterized by two-dimensional data. Each element in the matrix is identified by its position in the row and column format.In the given example, our matrix is:- \[ \begin{bmatrix} 2 & -5 & 1 \ -3 & 1 & 6 \ 4 & -2 & 3 \end{bmatrix} \]
This is a square matrix, meaning the number of rows and columns are equal, which is crucial for calculating its determinant. Working with a 3x3 matrix means interacting with 9 elements, forming its structure.One important aspect of this format is that linear transformations in a 3-dimensional space are easily represented by such matrices. This makes them particularly useful in graphics, engineering, physics, and more.

Determinant Formula

The determinant formula for a 3x3 matrix is a specific algebraic tool that gives a single scalar value, important in linear algebra. This value can help determine the properties of a matrix, including whether it is invertible (non-singular) or not.The formula for the determinant of a 3x3 matrix - \[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} \] is calculated as follows:- \[ \text{det}(A) = a_{11}(a_{22} \cdot a_{33} - a_{23} \cdot a_{32}) - a_{12}(a_{21} \cdot a_{33} - a_{23} \cdot a_{31}) + a_{13}(a_{21} \cdot a_{32} - a_{22} \cdot a_{31}) \]The formula combines the core idea of recursion and expansion, allowing one to break down the calculation into simpler two-dimensional arrays, often referred to as minors, that involve straightforward multiplication and subtraction.

Linear Algebra

Linear algebra is a vast field of mathematics that deals with vectors, matrices, and linear transformations. It forms the backbone of various applications in computer science, physics, and statistics. One of the primary elements in linear algebra is the matrix, used to describe linear mappings between different vector spaces. Understanding linear algebra involves mastering certain concepts and operations that simplify solving complex problems: - **Vectors:** Directed quantities with both magnitude and direction, depicted as points or arrows in space. - **Matrices:** Collections of numbers arranged in rows and columns, utilized to simplify calculations and model data relations. - **Determinants:** Provide insights into the properties of transformations and are critical in equation solving. Linear algebra also underlies algorithms in machine learning, optimization, and numerical simulations, making proficiency in this area highly beneficial for modern scientific challenges.

Matrix Operations

Matrix operations are fundamental tasks in linear algebra that include addition, subtraction, scalar multiplication, and multiplication of matrices. These operations allow us to manipulate matrices to solve equations or transform data. Here are the key operations:

• **Addition/Subtraction:** Performed element-wise, only possible between matrices of the same size.


• **Scalar Multiplication:** Every element of the matrix is multiplied by a number (scalar), effectively scaling the matrix.


• **Matrix Multiplication:** A more complex process that involves summing the products of elements from rows of the first matrix with columns of the second matrix. This is only possible when the number of columns in the first matrix equals the number of rows in the second matrix.

Each operation has a specific role in transforming or combining data, often employed in systems of equations or to model physical phenomena. Understanding these operations enables easier navigation through the intricacies of matrices in practical applications.