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Chapter 4: Problem 7

Compute the determinants using cofactor expansion along any row or column that seems convenient. \(\left|\begin{array}{rrr}5 & 2 & 2 \\ -1 & 1 & 2 \\ 3 & 0 & 0\end{array}\right|\)

Short Answer

Expert verified
The determinant is 18.

Step by step solution

01

Select a Row or Column for Expansion

A convenient column for cofactor expansion is the third column because it contains two zeros, simplifying our calculations.
02

Set Up the Cofactor Expansion

Using the third column for the cofactor expansion, the determinant can be expressed as: \[ \det(A) = a_{13}C_{13} + a_{23}C_{23} + a_{33}C_{33} \]where:- \(a_{13} = 2\), \(a_{23} = 2\), \(a_{33} = 0\).
03

Calculate Cofactors

Compute each cofactor:- \(C_{13} = (-1)^{1+3} \times \det\left(\begin{array}{cc}-1 & 1 \ 3 & 0\end{array}\right) = +(0 + 3) = 3\).- \(C_{23} = (-1)^{2+3} \times \det\left(\begin{array}{cc}5 & 2 \ 3 & 0\end{array}\right) = - (0 - 6) = 6\).- \(C_{33} = (-1)^{3+3} \times \det\left(\begin{array}{cc}5 & 2 \ -1 & 1\end{array}\right) = +(5 - (-2)) = 7\).Since \(a_{33} = 0\), the term \(a_{33}C_{33}\) will cancel out.
04

Compute the Determinant

Substitute the known values into the expression for the determinant:\[det(A) = 2(3) + 2(6) + 0(7) = 6 + 12 + 0 = 18.\]

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Key Concepts

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

Cofactor Expansion
When working with determinants of matrices, one powerful technique is cofactor expansion, also known as Laplace expansion. This method allows you to expand a determinant along any row or column. Choosing a row or column with the most zeros simplifies calculations, as seen in our example.

For a matrix element located at the intersection of the row \(i\) and column \(j\), its cofactor is calculated by removing the \(i\)-th row and \(j\)-th column from the matrix and computing the determinant of the resulting smaller matrix. The cofactor is then adjusted by a sign factor of \((-1)^{i+j}\).

In the example provided, we used the third column for cofactor expansion as it had two zeros. This made the calculations more efficient. Each term in the expansion requires multiplying the element by its corresponding cofactor, summing up to find the determinant.
Matrix Algebra
Matrix algebra is an essential part of linear algebra, involving various operations with matrices. Matrices are rectangular arrays of numbers, and their operations go beyond simple arithmetic.

Determinants are a scalar value that can be calculated from a square matrix and carry information about the matrix, like whether it is invertible. The computation of determinants through cofactor expansions exemplifies the depth of mathematical operations in matrix algebra.

Knowledge of matrix algebra allows us to solve systems of linear equations, find areas and volumes, or understand transformations applied to coordinates. By employing techniques like cofactor expansion, one can unlock the many applications of matrices in various fields of science and engineering.
Linear Algebra
Linear algebra studies vectors, vector spaces, and linear transformations, forming the backbone of many mathematical applications. It provides us with tools to analyze mathematical models that involve linear relationships.

In linear algebra, matrices serve a central role. They can represent linear transformations, systems of equations, and more. The calculation of a determinant as demonstrated in our task, is essential in understanding the properties of these transformations.

Linear algebra also finds its place in machine learning, computer graphics, and other modern day technologies, showcasing its importance beyond theoretical mathematics. Understanding concepts like cofactor expansion in determinants aids in utilizing these tools practically in various technological fields.

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Most popular questions from this chapter

The given matrix is of the form \(A=\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right] .\) In each case, \(A\) can be factored as the product of a scaling matrix and a rotation matrix. Find the scaling factor r and the angle \(\theta\) of rotation. Sketch the first four points of the trajectory for the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 1\end{array}\right]\) and classify the origin as a spiral attractor, spiral repeller, or orbital center. $$A=\left[\begin{array}{cc} 0 & 0.5 \\ -0.5 & 0 \end{array}\right]$$

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Many species of seal have suffered from commercial hunting. They have been killed for their skin, blubber, and meat. The fur trade, in particular, reduced some seal populations to the point of extinction. Today, the greatest threats to seal populations are decline of fish stocks due to overfishing, pollution, disturbance of habitat, entanglement in marine debris, and culling by fishery owners. Some seals have been declared endangered species; other species are carefully managed. Table 4.7 gives the birth and survival rates for the northern fur seal, divided into 2 -year age classes. [The data are based on A. E. York and J. R. Hartley, "Pup Production Following Harvest of Female Northern Fur Seals," Canadian Journal of Fisheries and Aquatic Science, \(38(1981),\) pp. \(84-90 .\) $$\begin{array}{ccc} \text { Age (years) } & \text { Birth Rate } & \text { Survival Rate } \\ \hline 0-2 & 0.00 & 0.91 \\ 2-4 & 0.02 & 0.88 \\ 4-6 & 0.70 & 0.85 \\ 6-8 & 1.53 & 0.80 \\ 8-10 & 1.67 & 0.74 \\ 10-12 & 1.65 & 0.67 \\ 12-14 & 1.56 & 0.59 \\ 14-16 & 1.45 & 0.49 \\ 16-18 & 1.22 & 0.38 \\ 18-20 & 0.91 & 0.27 \\ 20-22 & 0.70 & 0.17 \\ 22-24 & 0.22 & 0.15 \\ 24-26 & 0.00 & 0.00 \end{array}$$ (a) Construct the Leslie matrix \(L\) for these data and compute the positive eigenvalue and a corresponding positive eigenvector. (b) In the long run, what percentage of seals will be in each age class and what will the growth rate be?

Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of \(t .\) $$\begin{array}{l} x^{\prime}=\quad y-z, \quad x(0)=1 \\ y^{\prime}=x+\quad z, \quad y(0)=0 \\ z^{\prime}=x+y, \quad z(0)=-1 \end{array}$$
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