Question:
if P =$\begin{bmatrix}i&0&-i\\ 0&-i&i\\ -i&i&0\end{bmatrix}$ and $Q=\begin{bmatrix}-i&i\\ 0&0\\ i&-i\end{bmatrix}$ then $PQ$ is equal to
if P =$\begin{bmatrix}i&0&-i\\ 0&-i&i\\ -i&i&0\end{bmatrix}$ and $Q=\begin{bmatrix}-i&i\\ 0&0\\ i&-i\end{bmatrix}$ then $PQ$ is equal to
Updated On: Jul 6, 2022
- $\begin{bmatrix}-2&2\\ 1&-1\\ 1&-1\end{bmatrix}$
- $\begin{bmatrix}2&-2\\ -1&1\\ -1&1\end{bmatrix}$
- $\begin{bmatrix}2&-2\\ -1&1\\ \end{bmatrix}$
- $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
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The Correct Option is B
Solution and Explanation
Since, $P = \begin{bmatrix}i&1&-i\\ 0&-i&i\\ -i&i&0\end{bmatrix} and Q = \begin{bmatrix}-i&i\\ 0&0\\ i&-i\end{bmatrix}$
$\therefore PQ= \begin{bmatrix}i&0&-i\\ 0&-i&i\\ -i&i&0\end{bmatrix}\begin{bmatrix}-i&i\\ 0&0\\ i&-i\end{bmatrix}$
$= \begin{bmatrix}-i^{2}-i^{2}&i^{2}+i^{2}\\ i^{2}&-i^{2}\\ i^{2}&-i^{2}\end{bmatrix}\begin{bmatrix}1+1&-1-1\\ -1&1\\ -1&1\end{bmatrix}=\begin{bmatrix}2&-2\\ -1&1\\ -1&1\end{bmatrix}$
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.