Q32. - Algebra 2

Question

Compare the matrix used to reflect a figure over the x - axis to the matrix used to reflect a figure over the y -axis.

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

No, they are not inverses.

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

Step 1 ­- Description of step.

The matrix  A which is used to reflect a figure over the X -axis is given by:$A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

 

The matrix  B which is used to reflect a figure over the Y -axis is given by:$$" width="9" height="19" style="max-width: none; vertical-align: -4px;" >$B=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

 

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:

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

Therefore, it can be noticed that product of the matrices  A and  B is not equal to the identity matrix.

Therefore, the matrices A  and  B are not inverses of each other.

Step 3 ­- Description of step.

Therefore, no, they are not inverses.

b.

Step 1 ­- Description of step.

No, the matrices which are used to reflect the figure over the X -axis and Y -axis are not inverse of each other.

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

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

 $C=\left[\begin{array}{ccc}1& 2& 3\\ 2& 0& 1\end{array}\right]$

If the matrices which are used to reflect the figure over the X -axis and Y  -axis would have been inverse of each other, then after reflecting consecutively the figure over X -axis and  Y -axis, the same figure would have been obtained.

Step 2 ­- Description of step.

Reflect the triangle $\Delta ABC$  over the  X-axis by multiplying the matrix A  on the left with the vertex matrixC  .

Therefore, it can be obtained that the vertex matrix of the triangle $\Delta A\text{'}B\text{'}C\text{'}$  obtained after reflecting triangle $\Delta ABC$  over the X -axis is given by:

$\begin{array}{c}AC=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 2& 0& 1\end{array}\right]\\ =\left[\begin{array}{ccc}1+0& 2+0& 3+0\\ 0-2& 0-0& 0-1\end{array}\right]\\ =\left[\begin{array}{ccc}1& 2& 3\\ -2& 0& -1\end{array}\right]\end{array}$

Therefore, the vertex matrix of the triangle $\Delta A\text{'}B\text{'}C\text{'}$  obtained after reflecting triangle  $\Delta ABC$ over the  x-axis is: $\left[\begin{array}{ccc}1& 2& 3\\ -2& 0& -1\end{array}\right]$ .

Step 3 ­- Description of step.

Reflect the triangle $\Delta A\text{'}B\text{'}C\text{'}$  over the y -axis by multiplying the matrix  B on the left with the vertex matrixAC  .

Therefore, it can be obtained that the vertex matrix of the triangle  $\Delta A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}$ obtained after reflecting triangle  $\Delta A\text{'}B\text{'}C\text{'}$ over the  Y-axis is given by

$\begin{array}{c}BAC=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ -2& 0& -1\end{array}\right]\\ =\left[\begin{array}{ccc}-1+0& -2+0& -3+0\\ 0-2& 0+0& 0-1\end{array}\right]\\ =\left[\begin{array}{ccc}-1& -2& -3\\ -2& 0& -1\end{array}\right]\end{array}$

Therefore, the vertex matrix of the triangle $\Delta A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}$  obtained after reflecting triangle $\Delta A\text{'}B\text{'}C\text{'}$  over the  Y-axis is:$\left[\begin{array}{ccc}-1& -2& -3\\ -2& 0& -1\end{array}\right]$  .

Therefore, it can be noticed that the matrix  $\left[\begin{array}{ccc}-1& -2& -3\\ -2& 0& -1\end{array}\right]$ is not equal to the matrix$C=\left[\begin{array}{ccc}1& 2& 3\\ 2& 0& 1\end{array}\right]$  .

Therefore, same figure will not be obtained.

Therefore, they are not inverse of each other.

Step 4 ­- Description of step.

The graph showing the vertices of the triangles $\Delta ABC$ , $\Delta A\text{'}B\text{'}C\text{'}$  and  $\Delta A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}$ is:

Therefore, here also it can be noticed that the triangles $\Delta ABC$  is not same as that of $\Delta A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}$ .

Therefore, the matrices which are used to reflect the figure over theX  -axis and Y -axis are not inverse of each other.

Therefore, yes, our answer makes sense based on the geometry.