Math Problem Statement
Consider the matrix
š“=[ā3ā423]
Define š¶=š“ā1. Compute š“ā1 and give the following entries:
(1,2)-entry of š¶=š“ā1: š12= Answer 1 Question 5
(2,2)-entry of š¶=š“ā1: š22
Solution
We are given the matrix as:
-3 & -4 \\ 2 & 3 \end{pmatrix}$$ We need to compute the inverse of matrix $$A$$ (denoted $$C = A^{-1}$$) and find the following entries: - The $$(1, 2)$$-entry of $$C$$ (denoted $$c_{12}$$) - The $$(2, 2)$$-entry of $$C$$ (denoted $$c_{22}$$) ### Step 1: Compute the inverse of matrix $$A$$ For a 2x2 matrix $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, the inverse matrix is given by the formula: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ Where $$\text{det}(A) = ad - bc$$. For the given matrix: $$A = \begin{pmatrix} -3 & -4 \\ 2 & 3 \end{pmatrix}$$ - $$a = -3$$ - $$b = -4$$ - $$c = 2$$ - $$d = 3$$ Now, let's compute the determinant: $$\text{det}(A) = (-3)(3) - (-4)(2) = -9 + 8 = -1$$ ### Step 2: Compute the inverse matrix Now that we know the determinant is $$-1$$, we can compute the inverse: $$A^{-1} = \frac{1}{-1} \begin{pmatrix} 3 & 4 \\ -2 & -3 \end{pmatrix} = \begin{pmatrix} -3 & -4 \\ 2 & 3 \end{pmatrix}$$ Thus, the inverse of $$A$$ is: $$A^{-1} = \begin{pmatrix} -3 & -4 \\ 2 & 3 \end{pmatrix}$$ ### Step 3: Identify the requested entries - The $$(1, 2)$$-entry of $$A^{-1}$$ (denoted $$c_{12}$$) is $$-4$$. - The $$(2, 2)$$-entry of $$A^{-1}$$ (denoted $$c_{22}$$) is $$3$$. ### Final Answers: - $$c_{12} = -4$$ - $$c_{22} = 3$$ Would you like more details on any of the steps, or do you have additional questions? Here are some related questions you might find interesting: 1. How do you compute the determinant of a 2x2 matrix? 2. What happens when the determinant of a matrix is zero? 3. Can this method of computing the inverse be generalized to larger matrices? 4. How would you use the inverse matrix in solving systems of linear equations? 5. How do the entries of the inverse matrix relate to the original matrix? **Tip:** When calculating the inverse of a 2x2 matrix, always double-check the sign of the determinant, as it determines if the matrix is invertible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Inversion
Determinants
2x2 Matrices
Formulas
A^{-1} = (1/det(A)) * [d, -b; -c, a]
det(A) = ad - bc
Theorems
Inverse of a 2x2 matrix
Suitable Grade Level
Grades 10-12