Math Problem Statement

Consider the matrix

š“=[āˆ’3āˆ’423]

Define š¶=š“āˆ’1. Compute š“āˆ’1 and give the following entries:

(1,2)-entry of š¶=š“āˆ’1: š‘12= Answer 1 Question 5

(2,2)-entry of š¶=š“āˆ’1: š‘22

Solution

We are given the matrix AA 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.

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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