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Chapter 3: Problem 36

$$ \left[\begin{array}{ll} 0.1 & -3.2 \\ 0.1 & -1.5 \end{array}\right] $$

Short Answer

Expert verified
The inverse matrix of the given matrix A is: \(A^{-1} = \begin{bmatrix} -8.82352941 & 18.82352941 \\ -0.58823529 & 0.58823529 \end{bmatrix}\)

Step by step solution

01

Find the determinant of the matrix

To ensure the matrix is invertible, we need to calculate the determinant. For a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), the determinant is calculated as \(\det(A) = ad - bc\). In our case, let A = \[\begin{bmatrix} 0.1 & -3.2 \\ 0.1 & -1.5 \end{bmatrix}\]. Calculate the determinant: \(\det(A) = (0.1)(-1.5) - (-3.2)(0.1) = -0.15 + 0.32 = 0.17\) Since the determinant is non-zero, the given matrix is invertible.
02

Find the inverse matrix

To find the inverse of a 2x2 matrix, we can use the formula: \(A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\) We already have the determinant, so we can plug it into the formula: \(A^{-1} = \frac{1}{0.17} \begin{bmatrix} -1.5 & 3.2 \\ -0.1 & 0.1 \end{bmatrix}\) Now, we need to multiply each element in the matrix by the scalar \(\frac{1}{0.17}\).
03

Multiply by the scalar (1/ determinant)

We need to multiply each element in the matrix by the scalar to get the inverse matrix: \(A^{-1} = \begin{bmatrix} -1.5/0.17 & 3.2/0.17 \\ -0.1/0.17 & 0.1/0.17 \end{bmatrix} = \begin{bmatrix} -8.82352941 & 18.82352941 \\ -0.58823529 & 0.58823529 \end{bmatrix}\) So, the inverse matrix of the given matrix A is: \(A^{-1} = \begin{bmatrix} -8.82352941 & 18.82352941 \\ -0.58823529 & 0.58823529 \end{bmatrix}\)

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Key Concepts

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

Determinant Calculation
Finding the determinant of a matrix is an essential step in understanding whether a matrix is invertible. For a 2x2 matrix setup \[\begin{bmatrix} a & b \ c & d \end{bmatrix},\] the determinant is calculated using the formula:
\[\text{det}(A) = ad - bc.\]The determinant gives a scalar value that reveals key properties about the matrix. Specifically, if the determinant is zero, the matrix is singular, meaning it does not have an inverse. Conversely, a non-zero determinant indicates that the matrix is invertible. Calculating this scalar helps students determine the usability of the matrix in various mathematical operations and equations.
2x2 Matrix
A 2x2 matrix is one of the simplest forms of matrices, consisting of two rows and two columns. These matrices lay the foundation for more complex matrix operations and help illustrate fundamental concepts. When working with a 2x2 matrix:
  • The first element \(a_{11}\) is located at the first row and the first column.
  • The second element \(a_{12}\) is located at the first row and the second column.
  • The third element \(a_{21}\) is located at the second row and the first column.
  • The fourth element \(a_{22}\) is located at the second row and the second column.
Understanding this array structure helps in applying formulas like determinant calculations and finding the inverse. The simplicity of 2x2 matrices makes them a great introductory tool for learning about matrix algebra.
Invertible Matrix
An invertible matrix, also known as a non-singular matrix, is one that possesses an inverse. The concept of an inverse matrix is akin to finding a reciprocal in algebra. If a matrix \(A\) is invertible, there exists another matrix \(A^{-1}\) that satisfies the condition \[A \cdot A^{-1} = I,\] where \(I\) is the identity matrix of the same dimensions (2x2) which looks like \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}.\]For a matrix to be invertible, its determinant must not equal zero, which guarantees the unique transformation offered by the matrix can be reversed. In practical applications, knowing how to find the inverse of a matrix allows for solving linear equations and understanding transformations in vector space.

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Most popular questions from this chapter

Evaluate the given expression. Take \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .\) $$ 2 A-B $$

Translate the given systems of equations into matrix form. \(x+y-z=8\) \(2 x+y+z=4\) \(\frac{3 x}{4}+\frac{z}{2}=1\)

Revenue Karen Sandberg, your competitor in Suburban State U's T-shirt market, has apparently been undercutting your prices and outperforming you in sales. Last week she sold 100 tie dye shirts for \(\$ 10\) each, 50 (low quality) Crew shirts at \(\$ 5\) apiece, and 70 Lacrosse T-shirts for \(\$ 8\) each. Use matrix operations to calculate her total revenue for the week.

If \(A\) and \(B\) are \(2 \times 3\) matrices and \(A=B\), what can you say about \(A-B ?\) Explain.

I Population Movement In 2006, the population of the United States, broken down by regions, was \(55.1\) million in the Northeast, \(66.2\) million in the Midwest, \(110.0\) million in the South, and \(70.0\) million in the West. \({ }^{19}\) The matrix \(P\) below shows the population movement during the period 2006 \(2007 .\) (Thus, \(98.92 \%\) of the population in the Northeast stayed there, while \(0.17 \%\) of the population in the Northeast moved to the Midwest, and so on.) \(\begin{array}{ccll}\text { To } & \text { To } & \text { To } & \text { To } \\\ \text { NE } & \text { MW } & \text { S } & \text { W }\end{array}\) \(P=\begin{aligned}&\text { From NE } \\\&\text { From MW } \\\&\text { From S } \\\&\text { From W }\end{aligned}\left[\begin{array}{llll}0.9892 & 0.0017 & 0.0073 & 0.0018 \\ 0.0010 & 0.9920 & 0.0048 & 0.0022 \\ 0.0018 & 0.0024 & 0.9934 & 0.0024 \\ 0.0008 & 0.0033 & 0.0045 & 0.9914\end{array}\right]\) Set up the 2006 population figures as a row vector. Assuming that these percentages also describe the population movements from 2005 to 2006 , show how matrix inversion and multiplication allow you to compute the population in each region in 2005 . (Round all answers to the nearest \(0.1\) million.)
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