Solveeit Logo
Login

Question

Mathematics Question on Matrices

if P =[i0−i 0−ii −ii0]\begin{bmatrix}i&0&-i\\\ 0&-i&i\\\ -i&i&0\end{bmatrix} and Q=[−ii 00 i−i]Q=\begin{bmatrix}-i&i\\\ 0&0\\\ i&-i\end{bmatrix} then PQPQ is equal to

A

[−22 1−1 1−1]\begin{bmatrix}-2&2\\\ 1&-1\\\ 1&-1\end{bmatrix}

B

[2−2 −11 −11]\begin{bmatrix}2&-2\\\ -1&1\\\ -1&1\end{bmatrix}

C

[2−2 −11 ]\begin{bmatrix}2&-2\\\ -1&1\\\ \end{bmatrix}

D

[100 010 001]\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix}

Answer

[2−2 −11 −11]\begin{bmatrix}2&-2\\\ -1&1\\\ -1&1\end{bmatrix}

Explanation

Solution

Since, P=[i1−i 0−ii −ii0]andQ=[−ii 00 i−i]P = \begin{bmatrix}i&1&-i\\\ 0&-i&i\\\ -i&i&0\end{bmatrix} and Q = \begin{bmatrix}-i&i\\\ 0&0\\\ i&-i\end{bmatrix} ∴PQ=[i0−i 0−ii −ii0][−ii 00 i−i]\therefore PQ= \begin{bmatrix}i&0&-i\\\ 0&-i&i\\\ -i&i&0\end{bmatrix}\begin{bmatrix}-i&i\\\ 0&0\\\ i&-i\end{bmatrix} =[−i2−i2i2+i2 i2−i2 i2−i2][1+1−1−1 −11 −11]=[2−2 −11 −11]= \begin{bmatrix}-i^{2}-i^{2}&i^{2}+i^{2}\\\ i^{2}&-i^{2}\\\ i^{2}&-i^{2}\end{bmatrix}\begin{bmatrix}1+1&-1-1\\\ -1&1\\\ -1&1\end{bmatrix}=\begin{bmatrix}2&-2\\\ -1&1\\\ -1&1\end{bmatrix}