Math Problem Statement
Diketahui matriks A = [[-5, -2], [2, 1]]. Tentukan invers matriks A!
Solution
The image you uploaded contains a matrix :
-5 & -2 \\ 2 & 1 \end{bmatrix}$$ The task is to find the inverse of matrix $$A$$. To find the inverse of a 2x2 matrix $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the formula is: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ Where $$\text{det}(A)$$ (the determinant of $$A$$) is calculated as: $$\text{det}(A) = ad - bc$$ Now, for the matrix $$A$$, we have $$a = -5$$, $$b = -2$$, $$c = 2$$, and $$d = 1$$. Let's calculate the determinant first: $$\text{det}(A) = (-5)(1) - (-2)(2) = -5 + 4 = -1$$ Since the determinant is $$-1$$, we can now calculate the inverse of $$A$$: $$A^{-1} = \frac{1}{-1} \begin{bmatrix} 1 & 2 \\ -2 & -5 \end{bmatrix} = \begin{bmatrix} -1 & -2 \\ 2 & 5 \end{bmatrix}$$ Thus, the inverse of the matrix $$A$$ is: $$A^{-1} = \begin{bmatrix} -1 & -2 \\ 2 & 5 \end{bmatrix}$$ Would you like further details on the process or any other related questions? Here are 5 related questions you might find helpful: 1. How do you find the determinant of a 3x3 matrix? 2. What is the importance of the determinant in finding the inverse of a matrix? 3. How can you verify if two matrices are inverses of each other? 4. What happens if the determinant of a matrix is zero? 5. How do you compute the inverse of larger matrices? **Tip**: Always check that the determinant of a matrix is non-zero before attempting to find its inverse. If the determinant is zero, the matrix does not have an inverse (it is singular).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Inversion
Determinants
Linear Algebra
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