Question

For the matrix [A] given below, the transpose is __________. 

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]

• A. \( \begin{bmatrix} 2 & 1 & 4 \\ 3 & 4 & 3 \\ 4 & 5 & 2 \end{bmatrix} \)
• B. \( \begin{bmatrix} 4 & 3 & 2 \\ 5 & 4 & 1 \\ 2 & 3 & 4 \end{bmatrix} \)
• C. \( \begin{bmatrix} 4 & 2 & 3 \\ 5 & 1 & 4 \\ 2 & 4 & 3 \end{bmatrix} \)
• D. \( \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \)

Hint:

To transpose a matrix, flip it over its main diagonal. This is useful in many areas of linear algebra, especially in symmetric and orthogonal matrices.

Answer 1

The Correct Option is A

The transpose of a matrix is obtained by interchanging its rows and columns. That is, the element at position \((i, j)\) in the original matrix becomes the element at position \((j, i)\) in the transposed matrix.

Given:
\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]
Now taking the transpose:

- First row \((2\ 3\ 4)\) becomes the first column.
- Second row \((1\ 4\ 5)\) becomes the second column.
- Third row \((4\ 3\ 2)\) becomes the third column.

So, \[ A^T = \begin{bmatrix} 2 & 1 & 4 \\ 3 & 4 & 3 \\ 4 & 5 & 2 \end{bmatrix} \]
Hence, the correct option is (A).