Math Problem Statement
Solution
Thank you for providing the image. The task involves finding the inverse of the matrices listed in the exercise. Let's solve one as an example and provide steps you can follow for the rest.
General Formula for Finding the Inverse of a Matrix:
For a matrix , its inverse exists if and is given by: where .
Let's solve problem 1:
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Compute the determinant:
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Verify . Since it is 1, the inverse exists.
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Apply the formula:
Would you like me to solve all matrices or a specific one? Alternatively, I can provide a step-by-step solution for a matrix of your choice.
Questions for Further Exploration:
- What happens if ?
- Can you determine when a matrix is singular?
- How do inverses apply to systems of equations?
- What is the relationship between a matrix and its adjoint when computing the inverse?
- How can we extend the inverse computation to matrices?
Tip:
Always double-check the determinant before proceeding with the inverse calculation to avoid errors!
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Math Problem Analysis
Mathematical Concepts
Matrix Inversion
Determinants
Linear Algebra
Formulas
Inverse of a 2x2 matrix: A^(-1) = (1/det(A)) * [[d, -b], [-c, a]]
Determinant of a 2x2 matrix: det(A) = ad - bc
Theorems
Non-singular matrix theorem: A matrix is invertible if and only if its determinant is non-zero.
Suitable Grade Level
Grades 10-12