Question

If \(= P = \left[ {\begin{array}{*{20}{c}} 1&2\\ 3&4 \end{array}} \right]\) and \(= Q = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right]\) then Q^T P^T is 
1. \(\left[ {\begin{array}{*{20}{c}} 1&2\\ 3&4 \end{array}} \right]\)
2. \(\left[ {\begin{array}{*{20}{c}} 2&4\\ 1&3 \end{array}} \right]\)
3. \(\left[ {\begin{array}{*{20}{c}} 2&1\\ 4&3 \end{array}} \right]\)
4. \(\left[ {\begin{array}{*{20}{c}} 1&3\\ 2&4 \end{array}} \right]\)

Answer 1

Correct Answer - Option 2 : \(\left[ {\begin{array}{*{20}{c}} 2&4\\ 1&3 \end{array}} \right]\)

Concept:

Transpose of a Matrix:

If A = [aij]m × n, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A, denoted by A′ or (AT). AT = [aji]n × m

Calculation:

\(PQ = \left[ {\begin{array}{*{20}{c}} 1&3 \\ 2&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&4 \\ 1&3 \end{array}} \right]\)

\({\left( {PQ} \right)^T} = \left[ {\begin{array}{*{20}{c}} 2&4 \\ 1&3 \end{array}} \right]\)

(PQ)^T = Q^TP^T

QTP^T = \(\left[ {\begin{array}{*{20}{c}} 2&4 \\ 1&3 \end{array}} \right]\)