A=[■(1&2&3@4&5&6)], A^t=?Question 6Answera.[■(1&3&5@2&4&6)]b.[■(1&4@2&5@3&6)]c.Noned.[■(1&2@3&4@5&6)]

1. Break Down the Problem

We need to find the transpose of the matrix $A$.

2. Relevant Concepts

The transpose of a matrix $A$, denoted $A^t$, is obtained by swapping its rows with columns. For a matrix $A$ defined as: $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ the first row becomes the first column in the transpose, and the second row becomes the second column.

3. Analysis and Detail

The original matrix $A$ has 2 rows and 3 columns.

• The first row of $A$ is $[1, 2, 3]$.
• The second row of $A$ is $[4, 5, 6]$.

When we take the transpose $A^t$:

• The first column of $A^t$ will be the first row of $A$, which is $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$.
• The second column of $A^t$ will be the second row of $A$, which is $\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$.

Thus, we have: $A^t = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$

4. Verify and Summarize

The result shows that the dimensions of the matrix have been swapped: the new matrix $A^t$ has 3 rows and 2 columns.

Final Answer

The transpose of matrix $A$ is: $A^t = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$ This corresponds to answer choice b.