Math Problem Statement
Given that π΄ = [ 1 π 2 2 1 2 2 2 π ], where π and π are constants. If |π΄| = β13, evaluate the determinant of matrix [ 4 1 2 2 π 2 4 2 π ] using determinant properties.
Solution
To solve this problem, we'll use properties of determinants. Here's a step-by-step approach:
Step 1: Determine the given matrix
The matrix is given as:
1 & a & 2 \\ 2 & 1 & 2 \\ 2 & 2 & b \end{bmatrix}$$ It is also given that $$|A| = -13$$. ### Step 2: Express the matrix $$B$$ The matrix $$B$$ that we need to find the determinant of is: $$B = \begin{bmatrix} 4 & 1 & 2 \\ 2 & a & 2 \\ 4 & 2 & b \end{bmatrix}$$ ### Step 3: Observe the relationship between matrices $$A$$ and $$B$$ Notice that matrix $$B$$ can be transformed into matrix $$A$$ by performing certain row operations: - **Row 1** of $$B$$ is obtained by multiplying **Row 1** of $$A$$ by 4. - **Row 2** of $$B$$ is identical to **Row 2** of $$A$$. - **Row 3** of $$B$$ is obtained by multiplying **Row 1** of $$A$$ by 2. ### Step 4: Apply the determinant property The determinant of a matrix changes in the following ways under these row operations: - **Scaling a row** by a scalar $$k$$ multiplies the determinant by $$k$$. Thus, for the matrix $$B$$: $$|B| = 4 \cdot 2 \cdot |A| = 8 \cdot |A|$$ ### Step 5: Substitute the given determinant of $$A$$ Given $$|A| = -13$$, we substitute this value into the equation: $$|B| = 8 \cdot (-13) = -104$$ ### Final Answer: The determinant of matrix $$B$$ is $$\boxed{-104}$$. Would you like more details or have any questions? Here are 5 related questions: 1. How would the determinant change if only one row of $$A$$ was scaled? 2. Can you explain how row operations generally affect the determinant? 3. What would happen if two rows of a matrix were swapped? 4. How does the determinant help in understanding the properties of a matrix? 5. Could the determinant be zero even if $$a$$ and $$b$$ are non-zero? **Tip:** Remember that the determinant of a matrix can give you insights into whether the matrix is invertible (non-zero determinant means invertible).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Matrix Operations
Formulas
Determinant properties under row operations
Theorems
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Suitable Grade Level
Advanced High School