Problem 13 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix is -125.

Step 1: Identify the Matrix

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

Step 2: Use the Formula for a 3x3 Determinant

The determinant of a 3x3 matrix \( A \) is given by the formula: \[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\] where the elements of the matrix are identified as follows: \(A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\)

Step 3: Assign Values to Variables

Assign the values from the matrix to the variables in the formula: \[a = 3, \, b = 1, \, c = -2, \] \[d = 4, \, e = 2, \, f = 5, \] \[g = -6, \, h = 3, \, i = -1\]

Step 4: Calculate Each Part of the Formula

Calculate each part of the determinant formula separately:- Calculate \(ei - fh\): \[(2)(-1) - (5)(3) = -2 - 15 = -17\]- Calculate \(di - fg\): \[(4)(-1) - (5)(-6) = -4 + 30 = 26\]- Calculate \(dh - eg\): \[(4)(3) - (2)(-6) = 12 + 12 = 24\]

Step 5: Substitute the Values into the Formula

Substitute the calculated values of \(ei-fh\), \(di-fg\), \(dh-eg\) back into the determinant formula:\[\text{det}(A) = 3(-17) - 1(26) + (-2)(24)\]

Step 6: Compute the Final Expression

Compute the values:- First term: \(3 \times -17 = -51\)- Second term: \(1 \times 26 = 26\)- Third term: \(-2 \times 24 = -48\)Add them together:\[-51 - 26 - 48 = -125\]

Step 7: Conclude the Calculation

The determinant of the matrix is calculated as \(-125\).

Key Concepts

3x3 Matrix

A 3x3 matrix is a type of matrix with three rows and three columns, making a total of nine elements. Matrices are fundamental in mathematics and have a wide array of applications, from solving systems of equations to representing transformations in space. Here, the matrix is given as: \[\begin{bmatrix} 3 & 1 & -2 \ 4 & 2 & 5 \ -6 & 3 & -1 \end{bmatrix}\] This matrix is structured with rows represented horizontally and columns listed vertically. Each element of the matrix is represented by a numerical value. In context, these numbers could signify a variety of things depending on what the matrix is being used for, such as coefficients in a system of equations. Understanding this layout is crucial because it establishes the foundation for performing operations and determining properties such as the matrix's determinant.

Matrix Algebra

Matrix algebra is a branch of mathematics focusing on the study of matrices and their operations. It includes tasks like addition, subtraction, multiplication, and determining the determinant. Understanding the operations of matrices is crucial because they are used to solve complex problems in different areas like computer science, physics, and statistics. In the context of finding a determinant, it's essential to be comfortable with element indexing within the matrix and applying specific operations systematically.

• **Addition/Subtraction**: Involves adding or subtracting corresponding elements of matrices of the same size.
• **Multiplication**: Entails multiplying matrices by transforming rows of one matrix onto columns of another.
• **Determinant**: A special number that can be calculated from a square matrix; critical for solving equations and understanding matrix properties.

In this exercise, matrix algebra is utilized to calculate the determinant of the 3x3 matrix, an essential concept in solving matrix-related equations and transformations.

Determinant Formula

The determinant formula for a 3x3 matrix is a specific method used to calculate a special number known as the determinant. This value provides useful information regarding the matrix, such as its invertibility or the nature of solutions in a system of linear equations. For a matrix \[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \], the determinant \( \text{det}(A) \) is calculated using the following formula: \[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\] Here, the task involves substituting the elements of the 3x3 matrix into this formula, as demonstrated in the step by step solution provided. This formula is efficient and powerful in matrix algebra, making it easier to find the determinant without manually conducting expansive calculations. By applying the determinant formula, one can quickly determine matrix characteristics crucial for more advanced mathematical operations.