If matrix C = [[4, 3], [6, 3]], what is the transpose of C?a.[[4, 3], [3, 6]]b.[[3, 4], [6, 3]]c.[[4, 6], [3, 4]]d.[[4, 6], [3, 3]]

Break Down the Problem

1. Identify the given matrix $C$.
2. Understand what the transpose of a matrix is.

Relevant Concepts

1. The transpose of a matrix $A$, denoted as $A^T$, is formed by swapping its rows and columns. If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then: $A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$

Analysis and Detail

1. Given matrix: $C = \begin{bmatrix} 4 & 3 \\ 6 & 3 \end{bmatrix}$
2. Applying the transpose operation:
   • The first row $[4, 3]$ becomes the first column.
   • The second row $[6, 3]$ becomes the second column.
3. Hence, the transpose $C^T$ is: $C^T = \begin{bmatrix} 4 & 6 \\ 3 & 3 \end{bmatrix}$

Verify and Summarize

• By re-evaluating the transpose, we confirm that we have correctly swapped the rows and columns.
• The resulting matrix from the transpose operation is indeed: $C^T = \begin{bmatrix} 4 & 6 \\ 3 & 3 \end{bmatrix}$

Final Answer

Thus, the correct option is: d. $\begin{bmatrix} 4 & 6 \\ 3 & 3 \end{bmatrix}$