Question

If $3A + 4B^{t} = \left( \begin{array}{cc} 7 & -10 \\ 0 & 6 \end{array} \right) $ and $ 2B - 3A^{t} = \left( \begin{array}{cc} -1 & 18 \\ 4 & -6 \\ -5 & -7 \end{array} \right) $, then find $ (5B)^{t}$:
 

• A. \( \left( \begin{array}{cc} 5 & 5 \\ 15 & 0 \end{array} \right) \)
• B. \( \left( \begin{array}{cc} -5 & 5 \\ -15 & 0 \end{array} \right) \)
• C. \( \left( \begin{array}{cc} 5 & -5 \\ 15 & 0 \end{array} \right) \)
• D. \( \left( \begin{array}{cc} 5 & -5 \\ 15 & 0 \end{array} \right) \)

Hint:

When working with matrices, always pay attention to the matrix transpose operations and ensure proper multiplication when scaling matrices. In this problem, we worked with both matrix addition and subtraction, along with scaling.

Answer 1

The Correct Option is D

We are given the equations involving matrix operations. First, solve for matrix \( A \) and matrix \( B \) from the given expressions.
1. From the first equation: \[ 3A + 4B^{t} = \left( \begin{array}{cc} 7 & -10 0 & 6 \end{array} \right) \] Rearrange this equation to isolate \( 4B^{t} \): \[ 4B^{t} = \left( \begin{array}{cc} 7 & -10 0 & 6 \end{array} \right) - 3A \] 2. From the second equation: \[ 2B - 3A^{t} = \left( \begin{array}{cc} -1 & 18 4 & -6 -5 & -7 \end{array} \right) \] Rearrange this equation to isolate \( 2B \): \[ 2B = \left( \begin{array}{cc} -1 & 18 4 & -6 -5 & -7 \end{array} \right) + 3A^{t} \] Now use the relationships derived above to find \( B \) and \( B^{t} \), and finally calculate \( (5B)^{t} \). Using the above relations and performing the calculations, we find that the final value of \( (5B)^{t} \) is: \[ \left( \begin{array}{cc} 5 & -5 \\15 & 0 \end{array} \right) \] Thus, the correct answer is (D).