Let \( P=\left[\begin{array}{ll}6 & -1 \\ 2 & -4\end{array}\right] \) and \( M=\left[\begin{array}{cc}-6 & 2 \\ 2 & -7\end{array}\right] \) Calculate \( (P M)^{T} \) without calculatins \( P M \)

To find \( (PM)^T \) without directly calculating \( PM \), we can use the property of transposes that states: \[ (AB)^T = B^T A^T \] for any matrices \( A \) and \( B \). In our case, we have \( A = P \) and \( B = M \). Therefore, we can express \( (PM)^T \) as: \[ (PM)^T = M^T P^T \] Now, we need to calculate \( M^T \) and \( P^T \). 1. **Calculate \( P^T \)**: \[ P = \begin{bmatrix} 6 & -1 \\ 2 & -4 \end{bmatrix} \] The transpose \( P^T \) is obtained by swapping rows and columns: \[ P^T = \begin{bmatrix} 6 & 2 \\ -1 & -4 \end{bmatrix} \] 2. **Calculate \( M^T \)**: \[ M = \begin{bmatrix} -6 & 2 \\ 2 & -7 \end{bmatrix} \] The transpose \( M^T \) is: \[ M^T = \begin{bmatrix} -6 & 2 \\ 2 & -7 \end{bmatrix} \] Now we can substitute these transposes back into our expression for \( (PM)^T \): \[ (PM)^T = M^T P^T = \begin{bmatrix} -6 & 2 \\ 2 & -7 \end{bmatrix} \begin{bmatrix} 6 & 2 \\ -1 & -4 \end{bmatrix} \] Next, we will perform the matrix multiplication \( M^T P^T \): \[ M^T P^T = \begin{bmatrix} -6 & 2 \\ 2 & -7 \end{bmatrix} \begin{bmatrix} 6 & 2 \\ -1 & -4 \end{bmatrix} \] Calculating the elements of the resulting matrix: - First row, first column: \[ (-6)(6) + (2)(-1) = -36 - 2 = -38 \] - First row, second column: \[ (-6)(2) + (2)(-4) = -12 - 8 = -20 \] - Second row, first column: \[ (2)(6) + (-7)(-1) = 12 + 7 = 19 \] - Second row, second column: \[ (2)(2) + (-7)(-4) = 4 + 28 = 32 \] Putting it all together, we have: \[ (PM)^T = \begin{bmatrix} -38 & -20 \\ 19 & 32 \end{bmatrix} \] Thus, the final result is: \[ \boxed{\begin{bmatrix} -38 & -20 \\ 19 & 32 \end{bmatrix}} \]