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Chapter 8: Problem 32

Use matrix inversion to solve the system of equations. $$\left\\{\begin{aligned}4 x-3 y &=1 \\\2 x-y &=-1\end{aligned}\right.$$

Short Answer

Expert verified
The solution to the given system of equations via matrix inversion is \((x, y) = (-1,-1)\).

Step by step solution

01

Setting up the Matrix

First, let's re-write the system of equations in matrix form. A matrix \(A\) represented the coefficients and a vector \(V\) represents the constants on the right side of the equations. We get \[A= \begin{bmatrix} 4 & -3 \ 2 & -1 \end{bmatrix}\] and \[V= \begin{bmatrix} 1 \ -1 \end{bmatrix}\]
02

Finding the Inverse of Matrix

Next, calculate the inverse of the matrix \(A\) (if it exists). The inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(\frac{1}{{ad-bc}} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Using this formula, the inverse of our matrix \(A\) is \( \frac{1}{4*(-1)-(-3*2)} \begin{bmatrix} -1 & 3 \ -2 & 4 \end{bmatrix} = \begin{bmatrix} -1/2 & 3/2 \ -1 & 2 \end{bmatrix}\).
03

Solving the System

Finally, to solve for the vector of variables \(X = \begin{bmatrix} x \ y \end{bmatrix}\), multiply the inverse of matrix \(A\) by the vector \(V\). This gives us \[X=A^{-1}V = \begin{bmatrix}-1/2 & 3/2 \ -1 & 2 \end{bmatrix} \begin{bmatrix} 1 \ -1 \end{bmatrix} = \begin{bmatrix} -1 \ -1 \end{bmatrix}\]. Hence the solution to the system is \((x, y) = (-1, -1)\).

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Key Concepts

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

System of Linear Equations
A system of linear equations consists of two or more linear equations that share a set of variables. These equations relate the variables in such a way that all of them need to be true at the same time. Solving these systems is about finding the values for the variables that satisfy all equations simultaneously.

For example, consider the system:
\[\begin{align*}4x - 3y &= 1 \2x - y &= -1\end{align*}\]
This is a system of two equations with two variables, x and y. The objective is to find a pair \( (x, y) \) that makes both equations true. In this context, matrix inversion method comes into play as an efficient way to find this solution when the system can be represented in matrix form.
Matrix Algebra
Matrix algebra is a branch of mathematics that deals with the study of matrices and the ways in which they can be manipulated. Matrices are arrays of numbers or functions arranged in rows and columns that can represent a system of linear equations.

Operations in matrix algebra include addition, subtraction, multiplication, the calculation of the determinant, and finding the inverse of a matrix. For the matrix inversion method, multiplication and the inverse are particularly important. You would typically express a system of equations as a product of a matrix and a vector, where the matrix contains the coefficients of the variables, and the vector contains the constants. Using matrix multiplication, one can condense complex systems of linear equations into more manageable forms.
Inverse of a Matrix
The inverse of a matrix is analogous to the reciprocal of a number. Just as multiplying a number by its reciprocal gives 1, multiplying a square matrix by its inverse yields the identity matrix — which acts like 1 for matrices. The inverse of a matrix \(A\) is denoted as \(A^{-1}\).

However, not every matrix has an inverse. A matrix must be square (having the same number of rows and columns) and its determinant (a scalar value) must not be zero. The determinant provides information about the properties of the matrix, such as whether it has an inverse and if its rows and columns are linearly independent.

To find the inverse of a 2x2 matrix like
\[\begin{bmatrix}a & b \c & d \end{bmatrix}\]
the formula is:
\[A^{-1} = \frac{1}{{ad-bc}} \begin{bmatrix}d & -b \-c & a \end{bmatrix}\]
This formula changes if the matrix is larger than 2x2, but the principle remains the same. The inverse of a matrix is heavily used for solving a system of equations using matrix inversion method.

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Most popular questions from this chapter

Let \(A=\left[\begin{array}{ll}1 & -1 \\ 1 & -1\end{array}\right]\) and \(B=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] .\) What is the prod- uct \(A B ?\) Is it true that if \(A\) and \(B\) are matrices such that \(A B\) is defined and all the entries of \(A B\) are zero, then either all the entries of \(A\) must be zero or all the entries of \(B\) must be zero? Explain.

A couple has \(\$ 10,000\) to invest for their child's wedding. Their accountant recommends placing at least \(\$ 6000\) in a high-yield investment and no more than \(\$ 4000\) in a low-yield investment. (a) Use \(x\) to denote the amount of money placed into the high-yield investment. Use \(y\) to denote the amount of money placed into the low-yield investment. Write a system of linear inequalities that describes the possible amounts the couple could invest in each type of venture. (b) Graph the region that represents all possible amounts the couple could put into each investment if they wish to follow the accountant's advice.

Apply elementary row operations to a matrix to solve the system of equations. If there is no solution, state that the system is inconsistent. $$\left\\{\begin{array}{l}-x+2 y-3 z=2 \\ 2 x+3 y+2 z=1 \\ 3 x+y+5 z=1\end{array}\right.$$

Apply elementary row operations to a matrix to solve the system of equations. If there is no solution, state that the system is inconsistent. $$\left\\{\begin{array}{l}3 r+4 s-8 t=14 \\ 2 r-2 s+4 t=28\end{array}\right.$$

A chemist wishes to make 10 gallons of a \(15 \%\) acid solution by mixing a \(10 \%\) acid solution with a \(25 \%\) acid solution. (a) Let \(x\) and \(y\) denote the total volumes (in gallons) of the \(10 \%\) and \(25 \%\) solutions, respectively. Using the variables \(x\) and \(y,\) write an equation for the total volume of the \(15 \%\) solution (the mixture). (b) Using the variables \(x\) and \(y,\) write an equation for the total volume of acid in the mixture by noting that Volume of acid in \(15 \%\) solution \(=\) volume of acid in \(10 \%\) solution \(+\) volume of acid in \(25 \%\) solution. (c) Solve the system of equations from parts (a) and (b), and interpret your solution. (d) Is it possible to obtain a \(5 \%\) acid solution by mixing a \(10 \%\) solution with a \(25 \%\) solution? Explain without solving any equations.
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