Math Problem Statement
Find the inverse of the matrix
Solution
The matrix you've provided is:
1 & 1 \\ 1 & -2 \end{pmatrix}$$ To find the inverse of a 2x2 matrix, we use the following formula: $$\text{If } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ then } A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ For the given matrix: $$A = \begin{pmatrix} 1 & 1 \\ 1 & -2 \end{pmatrix}$$ Here, $$a = 1$$, $$b = 1$$, $$c = 1$$, and $$d = -2$$. First, we calculate the determinant $$\text{det}(A) = ad - bc$$: $$\text{det}(A) = (1)(-2) - (1)(1) = -2 - 1 = -3$$ Since the determinant is non-zero, the matrix is invertible. Now we can find the inverse: $$A^{-1} = \frac{1}{-3} \begin{pmatrix} -2 & -1 \\ -1 & 1 \end{pmatrix}$$ Simplifying further: $$A^{-1} = \begin{pmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{1}{3} \end{pmatrix}$$ So, the inverse of the matrix is: $$\boxed{A^{-1} = \begin{pmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{1}{3} \end{pmatrix}}$$ Would you like more details or have any questions? Here are some related questions to consider: 1. How is the determinant of a matrix calculated for larger matrices? 2. What happens if the determinant of a matrix is zero? 3. How can you verify that a matrix and its inverse multiply to give the identity matrix? 4. What are the applications of matrix inverses in solving systems of linear equations? 5. How would you find the inverse of a 3x3 matrix? **Tip:** When calculating the determinant of larger matrices, remember to break them down into smaller matrices using cofactor expansion.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Inverses of Matrices
Determinants
Formulas
Formula for the inverse of a 2x2 matrix
Theorems
Invertibility of 2x2 matrices
Suitable Grade Level
College Level