Math Problem Statement
Consider the following matrix A = [[4, 3, 1], [-2, -3, 0], [3, 2, 1]]. Choose the correct description of A. Find A^-1 if it exists.
Solution
The given matrix is:
4 & 3 & 1 \\ -2 & -3 & 0 \\ 3 & 2 & 1 \end{pmatrix}$$ ### 1. **Correct Description of Matrix $$A$$** To determine the correct description of the matrix, we need to check whether $$A$$ is invertible or singular (non-invertible). This can be done by calculating the determinant of the matrix. #### Determinant of $$A$$: $$\det(A) = 4 \begin{vmatrix} -3 & 0 \\ 2 & 1 \end{vmatrix} - 3 \begin{vmatrix} -2 & 0 \\ 3 & 1 \end{vmatrix} + 1 \begin{vmatrix} -2 & -3 \\ 3 & 2 \end{vmatrix}$$ Now, we calculate the minors: $$\begin{vmatrix} -3 & 0 \\ 2 & 1 \end{vmatrix} = (-3)(1) - (0)(2) = -3$$ $$\begin{vmatrix} -2 & 0 \\ 3 & 1 \end{vmatrix} = (-2)(1) - (0)(3) = -2$$ $$\begin{vmatrix} -2 & -3 \\ 3 & 2 \end{vmatrix} = (-2)(2) - (-3)(3) = -4 + 9 = 5$$ Thus, $$\det(A) = 4(-3) - 3(-2) + 1(5) = -12 + 6 + 5 = -1$$ Since the determinant of $$A$$ is non-zero ($$\det(A) = -1$$), the matrix is **invertible**. ### 2. **Finding $$A^{-1}$$** The inverse of a 3x3 matrix $$A$$ can be found using the formula: $$A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)$$ Since $$\det(A) = -1$$, the inverse of $$A$$ will be: $$A^{-1} = -\text{Adj}(A)$$ The adjugate (or adjoint) matrix, $$\text{Adj}(A)$$, is found by calculating the cofactor matrix and then transposing it. #### Step 1: Cofactor Matrix of $$A$$ The cofactor matrix $$C$$ is calculated by taking the determinant of the 2x2 submatrices obtained by deleting the corresponding row and column for each element of $$A$$. For $$C_{11}$$: $$\text{Cofactor of } A_{11} = \begin{vmatrix} -3 & 0 \\ 2 & 1 \end{vmatrix} = -3$$ For $$C_{12}$$: $$\text{Cofactor of } A_{12} = -\begin{vmatrix} -2 & 0 \\ 3 & 1 \end{vmatrix} = 2$$ For $$C_{13}$$: $$\text{Cofactor of } A_{13} = \begin{vmatrix} -2 & -3 \\ 3 & 2 \end{vmatrix} = 5$$ Following the same process for the rest of the entries, we get the cofactor matrix: $$C = \begin{pmatrix} -3 & 2 & 5 \\ 0 & -1 & 4 \\ 3 & -2 & -1 \end{pmatrix}$$ #### Step 2: Transpose of Cofactor Matrix The adjugate matrix $$\text{Adj}(A)$$ is the transpose of the cofactor matrix: $$\text{Adj}(A) = \begin{pmatrix} -3 & 0 & 3 \\ 2 & -1 & -2 \\ 5 & 4 & -1 \end{pmatrix}$$ #### Step 3: Multiply by $$\frac{1}{\det(A)}$$ Since $$\det(A) = -1$$, we multiply the adjugate matrix by $$-1$$: $$A^{-1} = -\text{Adj}(A) = \begin{pmatrix} 3 & 0 & -3 \\ -2 & 1 & 2 \\ -5 & -4 & 1 \end{pmatrix}$$ ### Final Answer: $$A^{-1} = \begin{pmatrix} 3 & 0 & -3 \\ -2 & 1 & 2 \\ -5 & -4 & 1 \end{pmatrix}$$ Would you like any further clarifications or details on the solution? ### Follow-up Questions: 1. What are the properties of invertible matrices in general? 2. How does the determinant relate to the invertibility of a matrix? 3. Can you find the inverse of a 2x2 matrix using the same method? 4. What happens if the determinant of a matrix is zero? 5. How is the adjugate matrix used in other applications besides finding the inverse? **Tip:** When solving matrix inversion problems, always check if the determinant is non-zero before proceeding with finding the inverse!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Determinants
Cofactors
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Inverse of a matrix: A^-1 = 1/det(A) * adjugate(A)
Cofactor matrix: Matrix of signed minors of A
Theorems
A matrix is invertible if and only if its determinant is non-zero.
The adjugate of a matrix is the transpose of its cofactor matrix.
Suitable Grade Level
University Level
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