Q33. - Algebra 2

Question

The matrix used to rotate a figure $\mathbf{270}\mathbf{°}$counter clockwise about the origin is $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$. Compare this matrix with the matrix used to rotate a figure counter clockwise about the origin.

1. Are they inverses?
2. Does your answer make sense based on the geometry? Use a drawing to support your answer.

Step-by-Step Solution

Answer

1. Yes, they are inverses.
2. Yes, our answer makes sense based on the geometry. The drawing to support our answer is:

Step 1 ­- Description of step.

The matrixA which is used to rotate a figure$270°$ counter clockwise about the origin is given by:$A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

The matrixB which is used to rotate a figure$90°$ counter clockwise about the origin is given by:$B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Step 2 ­- Description of step.

The two matrices $A$ and $B$ are said to be inverses if $AB=BA=I$, where $I$ is an identity matrix of the same order as that of$A$ and $B$.

Now, find out the product of the matrices $A$ and$B$.

Therefore, it can be obtained that:

$\begin{array}{c}AB=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\\ =\left[\begin{array}{cc}0+1& 0+0\\ 0+0& 1+0\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

Therefore, it can be noticed that product of the matrices $A$ and $B$ is equal to the identity matrix.

Therefore, the matrices $A$ and $B$ are inverses of each other.

Step 3 ­- Description of step.

Therefore, yes, they are inverses 

b

Step 1 ­- Description of step.

Yes, the matrices which are used to rotate a figure$270°$ counter clockwise about the origin and $90°$ counter clockwise about the origin are inverse of each other.

Consider the triangle $\Delta ABC$ having coordinates as $A\left(1,2\right)$, $B\left(2,0\right)$ and$C\left(3,1\right)$.

Therefore, the vertex matrix $C$ of the given triangle $\Delta ABC$ is given by: