Math Problem Statement
Find a general formula for the inverse of a 2 × 2 matrix A =
a b c d
Solution
The inverse of a 2x2 matrix can be found using the following formula:
For the matrix:
a & b \\ c & d \end{pmatrix}$$ The inverse of $$A$$, denoted as $$A^{-1}$$, is given by: $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ ### Explanation: 1. **Determinant**: The determinant of $$A$$ is $$\text{det}(A) = ad - bc$$. For the inverse to exist, the determinant must not be zero, i.e., $$ad - bc \neq 0$$. 2. **Matrix of Minors and Cofactors**: - Swap $$a$$ and $$d$$, - Negate $$b$$ and $$c$$. 3. **Adjugate (transpose of cofactors)**: This is the matrix $$\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$. 4. **Divide by the determinant**: The final inverse is obtained by dividing each element of the adjugate matrix by $$ad - bc$$. If the determinant $$ad - bc = 0$$, the matrix is **singular** and does not have an inverse. --- Do you want more details or have any questions? ### Related Questions: 1. How do you calculate the determinant of a 3x3 matrix? 2. Can you give an example of finding the inverse of a 2x2 matrix with specific values? 3. What happens if the determinant is zero in a 2x2 matrix? 4. How do you compute the inverse of larger matrices (3x3, 4x4, etc.)? 5. How is the inverse of a matrix used in solving systems of linear equations? ### Tip: Always check the determinant before trying to invert a matrix—if it's zero, the matrix has no inverse.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Determinants
Formulas
A^{-1} = (1 / (ad - bc)) * [[d, -b], [-c, a]]
Theorems
Inverse Matrix Theorem: A square matrix has an inverse if and only if its determinant is non-zero.
Suitable Grade Level
Undergraduate (or advanced high school mathematics)