Problem 30 (a) Express the system in the ma... [FREE SOLUTION]

Chapter 8: Problem 30

(a) Express the system in the matrix form $A X=B .$ (b) Approximate $A^{-1}$, using four-decimal-place accuracy for its elements. (c) Use $X=A^{-1} B$ to approximate the solution of the system to four-decimal-place accuracy. $$\left\\{\begin{array}{l} 1.9 x-3.2 y=5.7 \\ 2.6 x+0.4 y=3.8 \end{array}\right.$$

Short Answer

Expert verified

The solution is approximately $x = 1.6600$ and $y = -0.8605$.

Step by step solution

Identify Coefficients

First, identify the coefficients of the variables from each equation in the system. For the first equation $1.9x - 3.2y = 5.7$, the coefficients are $1.9$ and $-3.2$. For the second equation $2.6x + 0.4y = 3.8$, the coefficients are $2.6$ and $0.4$.

Express System as Matrices

Write the system of equations in the form $A\cdot X = B$, where $A$ is the matrix of coefficients, $X$ is the column matrix of variables, and $B$ is the matrix of constants. We have:\[A = \begin{pmatrix} 1.9 & -3.2 \ 2.6 & 0.4 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 5.7 \ 3.8 \end{pmatrix}\]

Calculate A's Determinant

Calculate the determinant of matrix $A$. The determinant of a 2x2 matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$ is given by $ad - bc$. Substituting the values from matrix $A$, we have:\[\text{det}(A) = (1.9)(0.4) - (-3.2)(2.6) = 0.76 + 8.32 = 9.08\]

Find A's Inverse

Use the formula for the inverse of a 2x2 matrix, which is:\[A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\]Substitute the values and the calculated determinant:\[A^{-1} = \frac{1}{9.08} \begin{pmatrix} 0.4 & 3.2 \ -2.6 & 1.9 \end{pmatrix}\]Calculate each element to four-decimal-place accuracy:\[\begin{pmatrix} 0.0440 & 0.3526 \ -0.2865 & 0.2093 \end{pmatrix}\]

Compute X = A^{-1}B

Now, calculate $X = A^{-1} B$ using matrix multiplication:\[X = \begin{pmatrix} 0.0440 & 0.3526 \ -0.2865 & 0.2093 \end{pmatrix} \times \begin{pmatrix} 5.7 \ 3.8 \end{pmatrix}\]Find the resulting matrix:\[X = \begin{pmatrix} (0.0440 \times 5.7) + (0.3526 \times 3.8) \ (-0.2865 \times 5.7) + (0.2093 \times 3.8) \end{pmatrix}\]Calculating the values:\[X = \begin{pmatrix} 1.6600 \ -0.8605 \end{pmatrix}\]

Interpret the Results

The solution matrix $X$ represents the values of $x$ and $y$. Therefore, $x \approx 1.6600$ and $y \approx -0.8605$ to four decimal places.

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Key Concepts

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

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the given exercise, we have two equations related to two variables, namely $x$ and $y$. These equations can be represented in matrix form to facilitate easier calculation and solution.

The system given is: \[ \begin{align*} 1.9x - 3.2y & = 5.7, \ 2.6x + 0.4y & = 3.8. \end{align*} \]
To express this system in matrix terms, we create a coefficient matrix $A$, a variable matrix $X$, and a constant matrix $B$: \[ A = \begin{pmatrix}1.9 & -3.2 \ 2.6 & 0.4\end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 5.7 \ 3.8 \end{pmatrix}. \]

Expressing the system in this format allows the use of matrix algebra techniques for solving it.

Matrix Inverse

The inverse of a matrix is a crucial concept when solving systems of equations using matrix algebra, specifically when aiming to find the solution in the form of $X = A^{-1}B$. Not all matrices have inverses; only those with a non-zero determinant do.To compute the matrix inverse:

First, ensure the matrix is square (it has the same number of rows and columns).
The inverse of a 2x2 matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$ can be found using the formula:\[ A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}. \]
Using the given matrix $A = \begin{pmatrix} 1.9 & -3.2 \ 2.6 & 0.4 \end{pmatrix}$ with a determinant of 9.08, calculate the inverse as: \[ A^{-1} = \frac{1}{9.08} \begin{pmatrix} 0.4 & 3.2 \ -2.6 & 1.9 \end{pmatrix} = \begin{pmatrix} 0.0440 & 0.3526 \ -0.2865 & 0.2093 \end{pmatrix}. \]

Calculating this accurately to four decimal places ensures precise solutions to the system of equations.

Determinant Calculation

The determinant of a matrix tells us many things about the matrix, such as whether it has an inverse or not. For a 2x2 matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, its determinant is calculated as $ad - bc$.In our exercise:

Matrix $A$ is: \[ A = \begin{pmatrix} 1.9 & -3.2 \ 2.6 & 0.4 \end{pmatrix}. \]
The determinant is computed as follows: \[ \text{det}(A) = (1.9 \times 0.4) - (-3.2 \times 2.6) = 0.76 + 8.32 = 9.08. \]

This non-zero determinant confirms that matrix $A$ is invertible, allowing us to use its inverse in solving the system of equations. Additionally, understanding how to calculate determinants is foundational for assessing the solvability of such systems.

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Short Answer

Step by step solution

Identify Coefficients

Express System as Matrices

Calculate A's Determinant

Find A's Inverse

Compute X = A^{-1}B

Interpret the Results

Key Concepts

System of Equations

Matrix Inverse

Determinant Calculation

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