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Chapter 6: Problem 11

Find the cofactor of each element in the second row for each matrix. $$\left[\begin{array}{rrr}1 & 2 & -1 \\\2 & 3 & -2 \\\\-1 & 4 & 1\end{array}\right]$$

Short Answer

Expert verified
The cofactors are -6, 0, and -6.

Step by step solution

01

Identify the Elements of the Second Row

The second row of the matrix is \([2, 3, -2]\). We will find the cofactor for each of these elements.
02

Construct the Minor for Element 2

Remove the second row and first column to construct the minor matrix for the element in position (2,1), which is \(3\times3\): \[ \begin{bmatrix} 2 & -1 \ 4 & 1 \end{bmatrix} \].
03

Calculate the Determinant of the Minor for Element 2

The determinant of \( \begin{bmatrix} 2 & -1 \ 4 & 1 \end{bmatrix} \) is calculated as \((2)(1) - (-1)(4) = 2 + 4 = 6\).
04

Find the Cofactor for Element 2

The cofactor is given by \((-1)^{2+1} \times 6 = -6\), applying the position-based sign change for the element (2,1).
05

Construct the Minor for Element 3

Remove the second row and second column to construct the minor matrix for the element in position (2,2), which is \(3\times3\): \[ \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} \].
06

Calculate the Determinant of the Minor for Element 3

The determinant of \( \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} \) is calculated as \((1)(1) - (-1)(-1) = 1 - 1 = 0\).
07

Find the Cofactor for Element 3

The cofactor is given by \((-1)^{2+2} \times 0 = 0\), applying the position-based sign change for the element (2,2).
08

Construct the Minor for Element -2

Remove the second row and third column to construct the minor matrix for the element in position (2,3), which is \(3\times3\): \[ \begin{bmatrix} 1 & 2 \ -1 & 4 \end{bmatrix} \].
09

Calculate the Determinant of the Minor for Element -2

The determinant of \( \begin{bmatrix} 1 & 2 \ -1 & 4 \end{bmatrix} \) is calculated as \((1)(4) - (2)(-1) = 4 + 2 = 6\).
10

Find the Cofactor for Element -2

The cofactor is given by \((-1)^{2+3} \times 6 = -6\), applying the position-based sign change for the element (2,3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Matrices
A matrix is a mathematical representation composed of numbers arranged in rows and columns. It is a fundamental concept in linear algebra. Matrices help solve systems of equations, transform geometric data, and more.
  • The size of a matrix is determined by its number of rows and columns, often denoted as 'm x n'.
  • A matrix with the same number of rows and columns is called a square matrix.
  • Elements in a matrix are typically denoted as \(a_{ij}\), where 'i' is the row number and 'j' is the column number.
Being comfortable with matrices is crucial as they are a foundational component in many mathematical applications.
Exploring the Concept of Minors
In matrix theory, a minor of an element in a matrix is obtained by eliminating the row and column of the element from the matrix. This is a smaller matrix formed from the original one. Calculating minors is a crucial step in finding determinants and cofactors.
  • To find the minor of element \(a_{ij}\), remove the i-th row and j-th column.
  • The minor is often used to simplify processes like finding the determinant of larger matrices.
  • Minors are also instrumental in the calculation of a matrix's inverse.
The concept of minors facilitates breaking down complex matrices into more manageable forms.
Calculating Determinants
Determinants provide a scalar value which is a function of a matrix. They are pivotal in understanding the properties of a matrix and solve linear systems. The determinant mainly applies to square matrices.
  • The determinant of a 2x2 matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] is calculated as \(ad - bc\).
  • For larger matrices, determinants are calculated recursively using minors and cofactors.
  • Determinants are useful, for example, in finding out if a system has a unique solution (non-zero determinant) or if matrices are invertible.
Understanding how to compute determinants is essential for various applications in algebra.
Role of Algebra in Understanding Matrices
Algebra is the branch of mathematics that studies symbols and the rules for manipulating them. It provides a powerful framework for working with mathematical concepts like matrices. Algebraic approaches help in devising systematic methods to perform operations on matrices.
  • Algebraic expressions allow us to define and operate on matrices in scalable ways.
  • Matrix manipulation, such as addition, subtraction, and multiplication, derives from algebraic principles.
  • Algebra is used to derive theorems and algorithms applied in matrix calculations, including determinants and inverses.
Mastery of algebraic methods enhances one's ability to navigate the complexities of matrix-related problems.

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Most popular questions from this chapter

Solve each application. Scheduling Production Caltek Computer Company makes two products: computer monitors and printers. Both require time on two machines: monitors, 1 hour on machine \(A\) and 2 hours on machine \(B\); printers, 3 hours on machine \(A\) and 1 hour on machine \(B\). Both machines operate 15 hours per day. What is the maximum number of each product that can be produced per day under these conditions?

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Graph the solution set of each system of inequalities by hand. $$\begin{array}{l} -2 < x < 3 \\ -1 \leq y \leq 5 \\ 2 x+y < 6 \end{array}$$

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned} &y=x^{2}-x+1\\\ &y=-x^{2}+1 \end{aligned}$$

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+B=B+A\) (commutative property)
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