Problem 18 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{rrrr} 2 & 5 & 1 & 0 \\ -4 & 0 & -3 & 0 \\ 3 & -2 & 1 & 6 \\ -1 & 4 & 2 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -378.

Step 1: Expand Along the Column with Zeros

Column 4 has three zeros (rows 1, 2, 4) and one non-zero entry (row 3: 6). Expanding along column 4:
\(\det = 6 \cdot (-1)^{3+4} \cdot M_{34}\) where \(M_{34}\) is the 3x3 minor.

Step 2: Calculate the Minor

\(M_{34} = \begin{vmatrix} 2 & 5 & 1 \\ -4 & 0 & -3 \\ -1 & 4 & 2 \end{vmatrix}\)
\(= 2(0+12) - 5(-8-3) + 1(-16-0)\)
\(= 2(12) - 5(-11) + 1(-16) = 24 + 55 - 16 = 63\)

Step 3: Compute the Determinant

\(\det = 6 \cdot (-1)^7 \cdot 63 = 6 \cdot (-1) \cdot 63 = -378\)

Key Concepts

Understanding a 4x4 Matrix

A 4x4 matrix is a collection of numbers arranged in four rows and four columns. This configuration forms a square matrix, which is important because square matrices allow for operations like finding the determinant. The determinant is a special number calculated from the elements of a square matrix, essentially giving us a tool to understand certain properties of the matrix.

When working with matrices, especially 4x4 ones, it's crucial to pay attention to the position of each element. Each position is identified using its row and column, like in a grid. For example, in the matrix provided, the element in the first row, first column, is 2, while the element in the second row, third column, is -3.

Understanding the layout of a 4x4 matrix helps us when we move on to find the determinant. Such matrices allow us to explore complex systems and solve various mathematical problems. They can be found in diverse fields like physics and computer science, proving their importance beyond just their numeric values alone.

Exploring Matrix Algebra

Matrix algebra is a powerful tool when dealing with matrices, and extends the basic operations you know from regular algebra to matrices. Operations like addition, subtraction, and multiplication can be performed on matrices, but only under certain conditions that relate to their size and configuration.

One of the most critical operations in matrix algebra is finding the determinant, especially for square matrices like the 4x4 matrix in our exercise. The determinant helps in assessing whether a matrix is invertible, which means whether its inverse exists. The inverse of a matrix is akin to division in regular algebra, helping solve equations that involve matrices.

• Addition and subtraction of matrices: They can only be done when the matrices are of the same order, meaning the same number of rows and columns.


• Multiplication of matrices: It involves the rows of the first matrix and the columns of the second matrix, sometimes leading to a different sized resultant matrix.


• Determinants: Useful for checking invertibility and other properties. For a 4x4 matrix, this involves more complex calculations, usually involving techniques like cofactor expansion.

These operations enable us to manipulate and analyze matrices effectively, providing a framework for solving real-world problems.

Step-by-Step Solution Approach

To solve the problem of finding the determinant of a 4x4 matrix, follow a systematic step-by-step approach. This approach ensures you don't get lost in the complex calculations involved.

Step 1: Identify the Matrix

This is crucial because only square matrices have determinants. Know the matrix as a 4x4 square matrix helps you understand the task at hand and recognize potential approaches, such as cofactor expansion.

Step 2: Choose a Row or Column for Expansion

For a 4x4 matrix, choosing a row or column with zeros might simplify calculations. The presence of zeros reduces the number of terms you need to consider, making it less intimidating.

Step 3: Perform Cofactor Expansion

Calculate the determinant using cofactor expansion by selecting a row or column and calculating the determinant of the resulting 3x3 matrices. This step involves recursive application of determinant calculation down to 2x2 matrices.

By breaking it down into these manageable parts, finding the determinant of a 4x4 matrix becomes more straightforward, providing you with insights to tackle similar problems in matrix algebra.