Question
Consider the following matrix: \[ A=\left(\begin{array}{ccc}1 & \cdots & 3 \\ \vdots & \ddots & \vdots \\ 4 & \cdots & 7\end{array}\right) \] It is known that: \[ |A|=19 \] Use the properties of matrices to determine the following: Let \( B \) be the matrix \( A \) with columns 1 and 3 interchanged. Let \( F \) be the matrix with row 2 multiplied by the scalar 4 .
Ask by Hills Adkins. in South Africa
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
- The determinant of matrix \( B \) is -19.
- The determinant of matrix \( F \) is 76.
Solution
1. The determinant of matrix \( A \) is given as
\[
|A| = 19.
\]
2. For matrix \( B \), columns 1 and 3 of \( A \) are interchanged. One of the basic properties of determinants is that swapping two columns (or rows) changes the sign of the determinant. Thus, we have:
\[
|B| = -|A| = -19.
\]
3. For matrix \( F \), the second row of \( A \) is multiplied by the scalar 4. A property of determinants states that multiplying one row by a scalar multiplies the determinant by that scalar. Therefore,
\[
|F| = 4|A| = 4 \times 19 = 76.
\]
Final Answers:
- \( |B| = -19 \)
- \( |F| = 76 \)
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Simplify this solution Mind Expander
When you interchange two columns of a matrix, the determinant changes its sign. Since \( |A| = 19 \), when you swap columns 1 and 3 of matrix \( A \) to form matrix \( B \), the determinant of \( B \) will be: \[ |B| = -|A| = -19. \] Now, if you take matrix \( A \) and multiply row 2 by a scalar \( 4 \) to get matrix \( F \), the determinant of the new matrix \( F \) is scaled by that same scalar. Therefore, the determinant of \( F \) is: \[ |F| = 4 \cdot |A| = 4 \cdot 19 = 76. \]
