Questions: Given A=[[-7, -3, -3, 8], [-3, -1, -5, 10]]. Find its transpose. [[8, -3, -3, -7], [10, -5, -1, -3]] [[-7, -3], [-3, -1], [-3, -5], [8, 10]] [[-8, -10], [3, 5], [3, 1], [7, 3]] [[8, 10], [-3, -5], [-3, -1], [-7, -3]]

Transcript text: Given $A=\left[\begin{array}{rrrr}-7 & -3 & -3 & 8 \\ -3 & -1 & -5 & 10\end{array}\right]$. Find its transpose. $\left[\begin{array}{rrrr}8 & -3 & -3 & -7 \\ 10 & -5 & -1 & -3\end{array}\right]$ $\left[\begin{array}{rr}-7 & -3 \\ -3 & -1 \\ -3 & -5 \\ 8 & 10\end{array}\right]$ $\left[\begin{array}{rr}-8 & -10 \\ 3 & 5 \\ 3 & 1 \\ 7 & 3\end{array}\right]$ $\left[\begin{array}{rr}8 & 10 \\ -3 & -5 \\ -3 & -1 \\ -7 & -3\end{array}\right]$

Solution

Solution Steps

To find the transpose of a matrix, we need to swap its rows with columns. For the given matrix $ A $, the first row becomes the first column, the second row becomes the second column, and so on.

Step 1: Define the Matrix

The given matrix $ A $ is defined as follows: \[ A = \begin{bmatrix} -7 & -3 & -3 & 8 \\ -3 & -1 & -5 & 10 \end{bmatrix} \]

Step 2: Calculate the Transpose

To find the transpose of matrix $ A $, we swap its rows and columns. The first row of $ A $ becomes the first column of $ A^T $, and the second row of $ A $ becomes the second column of $ A^T $. Thus, the transpose $ A^T $ is given by: \[ A^T = \begin{bmatrix} -7 & -3 \\ -3 & -1 \\ -3 & -5 \\ 8 & 10 \end{bmatrix} \]