Math Problem Statement

Obtain the inverse of the matrix (1 3, 2 -2) and hence solve the system: x + 3y = 4, 2x - 2y = 9.

Solution

The problem requires obtaining the inverse of the matrix

(1322)\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix}

and using it to solve the system of equations:

x+3y=4and2x2y=9x + 3y = 4 \quad \text{and} \quad 2x - 2y = 9

Step 1: Finding the Inverse of the Matrix

For a 2x2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}, the inverse is given by:

A1=1adbc(dbca)A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

In this case, the matrix is:

(1322)\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix}

So, we have:

  • a=1a = 1
  • b=3b = 3
  • c=2c = 2
  • d=2d = -2

The determinant det(A)=adbc\text{det}(A) = ad - bc is:

det(A)=(1)(2)(3)(2)=26=8\text{det}(A) = (1)(-2) - (3)(2) = -2 - 6 = -8

Now, the inverse of the matrix is:

A1=18(2321)=(14381418)A^{-1} = \frac{1}{-8} \begin{pmatrix} -2 & -3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix}

Step 2: Solving the System of Equations

We can express the system of equations as:

(1322)(xy)=(49)\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 9 \end{pmatrix}

Now, multiply both sides of the equation by the inverse matrix A1A^{-1}:

(xy)=A1(49)\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix}

First, compute:

A1(49)=(14381418)(49)A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix} = \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix} \begin{pmatrix} 4 \\ 9 \end{pmatrix}

This gives:

(14(4)+38(9)14(4)18(9))=(1+278198)=(35818)\begin{pmatrix} \frac{1}{4}(4) + \frac{3}{8}(9) \\ \frac{1}{4}(4) - \frac{1}{8}(9) \end{pmatrix} = \begin{pmatrix} 1 + \frac{27}{8} \\ 1 - \frac{9}{8} \end{pmatrix} = \begin{pmatrix} \frac{35}{8} \\ \frac{-1}{8} \end{pmatrix}

Thus, the solution is:

x=358,y=18x = \frac{35}{8}, \quad y = -\frac{1}{8}

Final Answer:

  • The inverse of the matrix is (14381418)\begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix}.
  • The solution to the system of equations is x=358x = \frac{35}{8} and y=18y = -\frac{1}{8}.

Would you like more details or have any questions?


Here are 5 related questions:

  1. What are the conditions for a 2x2 matrix to have an inverse?
  2. How do we calculate the determinant of a 3x3 matrix?
  3. Can a matrix with a determinant of zero have an inverse?
  4. How can you solve a system of equations using Gaussian elimination?
  5. How does Cramer's rule work in solving a system of equations?

Tip: The determinant of a matrix gives us key information about whether it is invertible. A determinant of zero means the matrix is singular and has no inverse.

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Math Problem Analysis

Mathematical Concepts

Matrix Inversion
Linear Equations
Determinants

Formulas

Inverse of a 2x2 matrix formula: A^{-1} = (1/(ad - bc)) * (d -b, -c a)
System of linear equations

Theorems

Matrix Inversion Theorem
Determinant Theorem

Suitable Grade Level

Grades 10-12