find the inverse of the matrix \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}.

\circ \quad \frac{1}{5} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}

\circ \quad \frac{1}{5} \begin{bmatrix} -3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3 \end{bmatrix}

Question

find the inverse of the matrix \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}.

\circ \quad \frac{1}{5} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}

\circ \quad \frac{1}{5} \begin{bmatrix} -3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3 \end{bmatrix}

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Matrix Determinant", "Cofactor Matrix", "Matrix Inverse Formula" ], "new_concepts": [], "current_concepts": [ "Matrix Determinant", "Cofactor Matrix", "Matrix Inverse Formula" ] } ## Calculate the determinant of the matrix Using the **Matrix Determinant** knowledge point $$ \begin{aligned} \det(A) &= 1\begin{vmatrix}1 & 2 \ 2 & 1\end{vmatrix} - 2\begin{vmatrix}2 & 2 \ 2 & 1\end{vmatrix} + 2\begin{vmatrix}2 & 1 \ 2 & 2\end{vmatrix}\ &= 1(1 - 4) - 2(2 - 4) + 2(4 - 2)\ &= -3 + 4 + 4 = 5 \end{aligned} $$ ## Find the cofactor matrix Using the **Cofactor Matrix** knowledge point $$ \begin{aligned} C_{11} &= (-1)^{1+1}\begin{vmatrix}1 & 2 \ 2 & 1\end{vmatrix} = -3, \quad C_{12} = (-1)^{1+2}\begin{vmatrix}2 & 2 \ 2 & 1\end{vmatrix} = 2, \quad C_{13} = (-1)^{1+3}\begin{vmatrix}2 & 1 \ 2 & 2\end{vmatrix} = 2\ C_{21} &= (-1)^{2+1}\begin{vmatrix}2 & 2 \ 2 & 1\end{vmatrix} = 2, \quad C_{22} = (-1)^{2+2}\begin{vmatrix}1 & 2 \ 2 & 1\end{vmatrix} = -3, \quad C_{23} = (-1)^{2+3}\begin{vmatrix}1 & 2 \ 2 & 2\end{vmatrix} = 2\ C_{31} &= (-1)^{3+1}\begin{vmatrix}2 & 2 \ 1 & 2\end{vmatrix} = 2, \quad C_{32} = (-1)^{3+2}\begin{vmatrix}1 & 2 \ 2 & 2\end{vmatrix} = 2, \quad C_{33} = (-1)^{3+3}\begin{vmatrix}1 & 2 \ 2 & 1\end{vmatrix} = -3 \end{aligned} $$ ## Construct the inverse matrix Using the **Matrix Inverse Formula** knowledge point $$ \begin{aligned} A^{-1} &= \frac{1}{\det(A)} C^T\ &= \frac{1}{5} \begin{bmatrix} -3 & 2 & 2 \ 2 & -3 & 2 \ 2 & 2 & -3 \end{bmatrix} \end{aligned} $$

(A) $\frac{1}{5} \begin{bmatrix} 1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1 \end{bmatrix}$

(B) $\frac{1}{5} \begin{bmatrix} -3 & 2 & 2 \ 2 & -3 & 2 \ 2 & 2 & -3 \end{bmatrix}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Matrix Inverse Formula" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the determinant of the matrix

Find the cofactor matrix

Construct the inverse matrix

Answer: