Math Problem Statement
Find the inverse of the given matrix: [ [2, 2], [2, 7] ].
Solution
To find the inverse of the given 2x2 matrix, we will follow these steps:
The matrix given is:
2 & 2 \\ 2 & 7 \end{pmatrix}$$ ### Step 1: Calculate the determinant. The formula for the determinant of a 2x2 matrix: $$\text{det}(A) = ad - bc$$ where the matrix is $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ For your matrix: $$a = 2, \quad b = 2, \quad c = 2, \quad d = 7$$ $$\text{det}(A) = (2)(7) - (2)(2) = 14 - 4 = 10$$ ### Step 2: Check if the determinant is zero. Since the determinant is 10 (which is not zero), the matrix is **invertible**. ### Step 3: Use the inverse formula for a 2x2 matrix. The inverse of a 2x2 matrix is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ Substituting the values: $$A^{-1} = \frac{1}{10} \begin{pmatrix} 7 & -2 \\ -2 & 2 \end{pmatrix}$$ Thus, the inverse is: $$A^{-1} = \begin{pmatrix} \frac{7}{10} & -\frac{2}{10} \\ -\frac{2}{10} & \frac{2}{10} \end{pmatrix}$$ or simplified: $$A^{-1} = \begin{pmatrix} 0.7 & -0.2 \\ -0.2 & 0.2 \end{pmatrix}$$ ### Final Answer: - Option **A** is correct. The inverse matrix is: $$\begin{pmatrix} 0.7 & -0.2 \\ -0.2 & 0.2 \end{pmatrix}$$ --- Would you like further details or have any questions? Here are five related questions to explore: 1. How do we compute the inverse of larger matrices (3x3 or more)? 2. What happens if the determinant of a matrix is zero? 3. Can every square matrix be inverted? 4. What are the practical applications of matrix inverses? 5. How does matrix multiplication work with inverse matrices? **Tip:** If a matrix has a zero determinant, it is called "singular" and does not have an inverse.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
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
Invertibility Theorem
Suitable Grade Level
Grades 10-12