Math Problem Statement
Use row reduction to find the inverse of the given matrix if it exists, and check your answer by multiplication. (If the inverse doesn't exist, enter DNE in any cell of the matrix.)
1 2
1 1
Solution
To find the inverse of the given matrix using row reduction, we augment the matrix with the identity matrix and then row reduce it to transform into . The steps are detailed below:
Step 1: Write the augmented matrix
1 & 2 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{array}\right]$$ ### Step 2: Perform row operations 1. Subtract Row 1 from Row 2 to make the entry in the $$(2,1)$$ position zero: $$R_2 \to R_2 - R_1$$ $$\left[\begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 0 & -1 & -1 & 1 \end{array}\right]$$ 2. Divide Row 2 by $$-1$$ to make the pivot in the $$(2,2)$$ position equal to 1: $$R_2 \to -R_2$$ $$\left[\begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 0 & 1 & 1 & -1 \end{array}\right]$$ 3. Subtract 2 times Row 2 from Row 1 to make the entry in the $$(1,2)$$ position zero: $$R_1 \to R_1 - 2R_2$$ $$\left[\begin{array}{cc|cc} 1 & 0 & -1 & 2 \\ 0 & 1 & 1 & -1 \end{array}\right]$$ The left side is now the identity matrix, and the right side is the inverse: $$A^{-1} = \begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix}$$ ### Step 3: Verify the inverse by multiplication To check if the computed matrix is correct, multiply $$A$$ and $$A^{-1}$$: $$A \cdot A^{-1} = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix}$$ Perform the multiplication: $$\begin{aligned} \text{Row 1, Column 1: } & (1)(-1) + (2)(1) = -1 + 2 = 1 \\ \text{Row 1, Column 2: } & (1)(2) + (2)(-1) = 2 - 2 = 0 \\ \text{Row 2, Column 1: } & (1)(-1) + (1)(1) = -1 + 1 = 0 \\ \text{Row 2, Column 2: } & (1)(2) + (1)(-1) = 2 - 1 = 1 \\ \end{aligned}$$ The product is: $$A \cdot A^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ Since the product is the identity matrix, the computed inverse is correct. ### Final Answer: $$A^{-1} = \begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix}$$ --- Would you like further details or help with a related topic? Here are some related questions for exploration: 1. What is the geometric significance of the inverse of a matrix? 2. How does the determinant of a matrix relate to its invertibility? 3. What are the computational differences between using row reduction and the adjoint method for finding inverses? 4. How does one find the inverse of larger matrices using LU decomposition? 5. What happens if the determinant of a matrix is zero? **Tip:** Always verify the inverse by checking if $$A \cdot A^{-1} = I$$ to confirm accuracy!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Row Reduction
Matrix Inversion
Formulas
Augmented matrix: [A|I] → [I|A⁻¹]
Inverse of a 2x2 matrix: A⁻¹ = 1/det(A) * adj(A)
Theorems
Invertibility Criterion (Non-zero determinant)
Suitable Grade Level
Grades 10-12
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