Math Problem Statement
Solution
The problem requires obtaining the inverse of the matrix
and using it to solve the system of equations:
Step 1: Finding the Inverse of the Matrix
For a 2x2 matrix , the inverse is given by:
In this case, the matrix is:
So, we have:
The determinant is:
Now, the inverse of the matrix is:
Step 2: Solving the System of Equations
We can express the system of equations as:
Now, multiply both sides of the equation by the inverse matrix :
First, compute:
This gives:
Thus, the solution is:
Final Answer:
- The inverse of the matrix is .
- The solution to the system of equations is and .
Would you like more details or have any questions?
Here are 5 related questions:
- What are the conditions for a 2x2 matrix to have an inverse?
- How do we calculate the determinant of a 3x3 matrix?
- Can a matrix with a determinant of zero have an inverse?
- How can you solve a system of equations using Gaussian elimination?
- 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
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