Question

If, A = \(\begin{bmatrix}a& d& l\\[0.3em]b& e& m\\[0.3em]c& f& n\\[0.3em] \end{bmatrix}\)and B = \(\begin{bmatrix}l& m& n\\[0.3em]a& b& c\\[0.3em]d& e& f\\[0.3em]  \end{bmatrix}\), then

• A. A = B
• B. A = 2B
• C. A = - B
• D. A = - 2B

Answer 1

The Correct Option is A

To determine the relationship between matrices \(A\) and \(B\), let's analyze both:

Matrix \(A\) is defined as:
\(\begin{bmatrix}a & d & l\\[0.3em]b & e & m\\[0.3em]c & f & n\\[0.3em]\end{bmatrix}\)

Matrix \(B\) is defined as:
\(\begin{bmatrix}l & m & n\\[0.3em]a & b & c\\[0.3em]d & e & f\\[0.3em]\end{bmatrix}\)

Observe that matrix \(B\) is the transpose of matrix \(A\). The transpose of a matrix \(A\), denoted by \(A^T\), involves swapping the rows and columns.

For matrix \(A\):
1. First row becomes first column: \([a, d, l]\rightarrow[l, a, d]\).
2. Second row becomes the second column: \([b, e, m]\rightarrow[m, b, e]\).
3. Third row becomes the third column: \([c, f, n]\rightarrow[n, c, f]\).

Thus, matrix \(B\) rearranges the rows of matrix \(A\) as its columns, confirming that \(B\) is indeed \(A^T\).

In terms of option analysis:

A = B

Since \(B\) is a direct transpose of \(A\), \(A\) is not equivalent to B without specific conditions like symmetry, which isn't stated here. Therefore:

Since \(A\) is not directly equal to \(B\) as a transpose does not confirm equivalence directly, none of the provided options directly reflect this transpose relationship. However, if internal consistency or misinterpretation exists, revisiting with symmetric conditions, not present here, can be re-evaluated.