Verify that (2A)T = 2AT
Given:
\( A = \begin{bmatrix} 2 & -3 \\ -7 & 5 \end{bmatrix} \)
To Verify:
\( (2A)^T = 2A^T \)
Step 1: Find 2A
\[ 2A = 2 \begin{bmatrix} 2 & -3 \\ -7 & 5 \end{bmatrix} = \begin{bmatrix} 4 & -6 \\ -14 & 10 \end{bmatrix} \]
Step 2: Find (2A)T
\[ (2A)^T = \begin{bmatrix} 4 & -14 \\ -6 & 10 \end{bmatrix} \]
Step 3: Find AT
\[ A^T = \begin{bmatrix} 2 & -7 \\ -3 & 5 \end{bmatrix} \]
Step 4: Find 2AT
\[ 2A^T = 2 \begin{bmatrix} 2 & -7 \\ -3 & 5 \end{bmatrix} = \begin{bmatrix} 4 & -14 \\ -6 & 10 \end{bmatrix} \]
Conclusion:
\[ (2A)^T = 2A^T = \begin{bmatrix} 4 & -14 \\ -6 & 10 \end{bmatrix} \]
Hence Verified.
Concept Used:
The transpose of a scalar multiple of a matrix equals the scalar multiple of the transpose:
\[ (kA)^T = kA^T \]