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Magazine Chapter 10: Problem 7
For Problems \(1-10\), use expansion by minors to evaluate each determinant. (Objective 1) $$ \left|\begin{array}{rrr} 3 & -4 & -2 \\ 5 & -2 & 1 \\ 1 & 0 & 0 \end{array}\right| $$
Short Answer
Expert verified
The determinant of the matrix is \(-8\).
Step by step solution
01
Understand the Matrix
We have a 3x3 matrix, and we need to find its determinant using the expansion by minors method. The matrix is:\[ \begin{pmatrix} 3 & -4 & -2 \ 5 & -2 & 1 \ 1 & 0 & 0 \end{pmatrix} \]
02
Select a Row or Column for Expansion
Choose the first row for expansion by minors (you can choose any row or column, but row 1 is stacked with some zeroes which might make the calculations easier).
03
Expansion by Minors
Calculate the determinant by expanding along the first row:\[ \text{Det} = 3 \cdot \text{minor}(1,1) + (-1)\cdot(-4) \cdot \text{minor}(1,2) + (-1)^2 \cdot (-2) \cdot \text{minor}(1,3) \]
04
Calculate Each Minor
1. For minor(1,1): Consider the 2x2 matrix \(\begin{pmatrix} -2 & 1 \ 0 & 0 \end{pmatrix}\). The determinant is \[ (-2)(0) - (1)(0) = 0 \].2. For minor(1,2): Consider the 2x2 matrix \(\begin{pmatrix} 5 & 1 \ 1 & 0 \end{pmatrix}\). The determinant is \[ (5)(0) - (1)(1) = -1 \].3. For minor(1,3): Consider the 2x2 matrix \(\begin{pmatrix} 5 & -2 \ 1 & 0 \end{pmatrix}\). The determinant is \[ (5)(0) - (-2)(1) = 2 \].
05
Substitute Minors Back Into the Expansion Equation
Substitute the calculated minors back into the expansion equation:\[ \text{Det} = 3 \cdot 0 + 4 \cdot (-1) + (-2) \cdot 2 \]
06
Simplify and Determine the Determinant
Compute each term and sum them:\[ \text{Det} = 0 + (-4) + (-4) = -8 \]
07
Verify the Result
Review each step to make sure that all calculations are correct. The expanded terms and sign adjustments seem correct. Thus, the determinant is \(-8\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Expansion by Minors
Expansion by minors is a method used to evaluate the determinant of a matrix. This technique involves a systematic approach to break down a larger matrix determinant into simpler ones. Here's how it works:
- Select a row or a column from your matrix (usually one with zeros or smaller numbers to simplify calculations).
- For each element in this row or column, calculate its corresponding minor. The minor of a particular matrix element is the determinant of the submatrix obtained by deleting the row and column containing that element.
- Apply the cofactor, which considers the position of the element, using the formula \((-1)^{i+j}\times\text{{det(minor)}}\), where \(i\) and \(j\) represent the row and column indices of the element.
- Sum up all these products (element \times\ cofactor) to find the determinant.
Matrix Determinant
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix, including:
- Whether the matrix is invertible: A non-zero determinant means the matrix has an inverse, while a zero determinant implies it doesn't.
- The volume distortion of linear transformations represented by the matrix: The absolute value of the determinant determines scale changes in volume.
- Geometric applications: In 2D, it can determine area scaling; in 3D, it relates to volume scaling.
3x3 Matrix
A 3x3 matrix is a grid of numbers with three rows and three columns. Understanding how to manipulate and evaluate this structure is crucial in linear algebra for students. Here's why it's important:
- It's the smallest square matrix that can represent complex transformations in three-dimensional space.
- They are often used to solve systems of linear equations.
- Knowing its determinant informs you if the matrix is invertible, which is essential for applications in engineering, graphics, and physics.
