Math Problem Statement
اجب بصح أو خط
The inverse of matrix A is provided. The question asks if the given matrix inverse is correct or incorrect: اجب بصح أو خط
Solution
The question asks whether the inverse of matrix is correctly given as:
3 & 2 & -1 \\ 1 & 6 & 3 \\ 2 & -4 & 0 \end{bmatrix}$$ and its inverse $$A^{-1}$$ as: $$A^{-1} = \frac{1}{64} \begin{bmatrix} 12 & 4 & 12 \\ 6 & 2 & -10 \\ -16 & 16 & 16 \end{bmatrix}$$ To check the correctness of the inverse matrix, we will: 1. Verify if the determinant of matrix $$A$$ is non-zero (since only matrices with non-zero determinant have an inverse). 2. Check if the provided inverse satisfies the property $$A \times A^{-1} = I$$, where $$I$$ is the identity matrix. Let me compute the determinant and confirm if the inverse is correct.The determinant of matrix $$A$$ is 64, which means that matrix $$A$$ is invertible (since the determinant is non-zero). Now, I will check if the given matrix is indeed the inverse by multiplying $$A$$ with the proposed inverse matrix and verifying if the result is the identity matrix.The product of $$A$$ and the given inverse matrix results in the identity matrix: $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ This confirms that the provided inverse is correct. Therefore, the answer to the question is **"صواب" (Correct)**. Would you like any further details? Here are five related questions to explore further: 1. How is the determinant of a 3x3 matrix calculated? 2. What are the conditions for a matrix to have an inverse? 3. How do you compute the inverse of a matrix manually? 4. What are some properties of the identity matrix? 5. Why is the determinant important in linear algebra? **Tip:** Always verify the determinant of a matrix before attempting to compute its inverse to ensure the matrix is invertible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Inversion
Determinants
Matrix Multiplication
Formulas
Matrix Inverse Formula: A^{-1} = (1/det(A)) * adj(A)
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
The Inverse Matrix Theorem
Properties of the Identity Matrix
Suitable Grade Level
Grades 11-12 or University Level
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