# Matrix Transformation

Which transformation does the matrix $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ represent?

Given, Transformation matrix (T.M.) = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
= $\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}$
= $\begin{bmatrix} -y \\ -x \end{bmatrix}$

So, the given transformation matrix represents a reflection on line x = -y.

Which transformation does the matrix $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ represent?

Given, Transformation matrix (T.M.) = $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
= $\begin{bmatrix} (-1)x + (0)y \\ (0)x + (1)y \end{bmatrix}$
= $\begin{bmatrix} -x \\ y \end{bmatrix}$

So, the given transformation matrix represents reflection on y-axis.

Which transformation does the matrix $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ represent? Using the matrix find the image of the point A(6, -2).

First part,
Given, Transformation matrix (T.M.) = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
= $\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}$
= $\begin{bmatrix} -y \\ -x \end{bmatrix}$

So, the given transformation matrix represents reflection on the line x = -y.

Second part,
A(6, -2) $\xrightarrow{\text{x = -y}}$ A'(2, -6)

Find the transformation matrix which represents the reflection on the line y = -x.

Let, Object = $\begin{bmatrix} x \\ y \end{bmatrix}$ and Transformation matrix (T.M.) = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
Image on reflection on y = -x is $\begin{bmatrix} -y \\ -x \end{bmatrix}$

Now, we know, Image = T.M. × Object or, $\begin{bmatrix} -y \\ -x \end{bmatrix}$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
or, $\begin{bmatrix} -y \\ -x \end{bmatrix}$ = $\begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$

Above condition will be true if,
a = 0, b = -1
c = -1, d = 0

So, Transformation matrix = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$

Find the transformation matrix which represents the reflection on Y-axis.

Let, Object = $\begin{bmatrix} x \\ y \end{bmatrix}$ and Transformation matrix (T.M.) = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
Image on reflection on y-axis is $\begin{bmatrix} -x \\ y \end{bmatrix}$

Now, we know, Image = T.M. × Object or, $\begin{bmatrix} -x \\ y \end{bmatrix}$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
or, $\begin{bmatrix} -x \\ y \end{bmatrix}$ = $\begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$

Above condition will be true if,
a = -1, b = 0
c = 0, d = 1

So, Transformation matrix = $\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}$

What transformation does the matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ represent? Use this matrix to find the image of a point (2, 4).

First part,
Given, Transformation matrix (T.M.) = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ × $\begin{bmatrix} x \ y \end{bmatrix}$
= $\begin{bmatrix} (1)x + (0)y \ (0)x + (1)y \end{bmatrix}$
= $\begin{bmatrix} x \ y \end{bmatrix}$

The value of object and image is same. So, the matrix represents identity transformation.

Second part,
A(2, 4) $\rightarrow$ A'(2, 4)

Find a 2 × 1 matrix which transforms a point (a, b) into the point (a + 2, b - 3). Using the same 2 ×1 matrix transform the point (5, 7).

Let 2 × 1 matrix be $\begin{pmatrix} p \\ q \end{pmatrix}$
So, (a, b) $\xrightarrow{T \begin{pmatrix} p \\ q \end{pmatrix}}$ (a + p, b + q)
Given, (a, b) $\rightarrow$ (a + 2, b - 3)

Comparing the corresponding value,
p = 2 and q = -3
So, 2 × 1 matrix = $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$

Now, using 2 ×1 matrix to transform the point (5, 7),
(5, 7) $\xrightarrow{T \begin{pmatrix} 2 \ -3 \end{pmatrix}}$ (5 + 2, 7 - 3) = (7, 4)

Find a 2 × 1 matrix which transforms a point (a, b) into the point (a + 4, b - 5). Using the same 2 ×1 matrix transform the point (-4, 6).

Let 2 × 1 matrix be $\begin{pmatrix} p \\ q \end{pmatrix}$
So, (a, b) $\xrightarrow{T \begin{pmatrix} p \\ q \end{pmatrix}}$ (a + p, b + q)
Given, (a, b) $\rightarrow$ (a + 4, b - 5)

Comparing the corresponding value,
p = 4 and q = -5
So, 2 × 1 matrix = $\begin{pmatrix} 4 \\ -5 \end{pmatrix}$

Now, using 2 ×1 matrix to transform the point (5, 7),
(-4, 6) $\xrightarrow{T \begin{pmatrix} 4 \ -5 \end{pmatrix}}$ (-4 + 4, 6 - 5) = (0, 1)

What transformation does the matrix $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ denote? Using the given matrix, find the image of the point A(5, -7).

First part,
Given, Transformation matrix (T.M.) = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
= $\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}$
= $\begin{bmatrix} -y \\ -x \end{bmatrix}$

So, the given transformation matrix denote reflection at x = -y.

Second part,
Object = A = $\begin{bmatrix} 5 \\ -7 \end{bmatrix}$

Now, We know, Image = T.M. × Object or, A' = $\begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} 5 \ -7 \end{bmatrix}$
so, A' = $\begin{bmatrix} 7 \ -5 \end{bmatrix}$

What transformation does the matrix $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ represent? Use the given matrix to transform the point (-4, 5).

First part,
Given, Transformation matrix (T.M.) = $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
= $\begin{bmatrix} (0)x + (1)y \\ (-1)x + (0)y \end{bmatrix}$
= $\begin{bmatrix} y \\ -x \end{bmatrix}$

So, the given transformation matrix represents rotation at -90° through origin.

Second part,
Object = A = $\begin{bmatrix} -4 \\ 5 \end{bmatrix}$

Now, We know, Image = T.M. × Object or, A' = $\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} -4 \ 5 \end{bmatrix}$
so, A' = $\begin{bmatrix} 5 \ 4 \end{bmatrix}$

If P(a, b) is transformed by $\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix}$ to P'(-10, -8), find the value of a and b.

Here, Object = P = $\begin{bmatrix} a \\ b \end{bmatrix}$
Transformation matrix (T.M.) = $\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix}$
Image = P' = $\begin{bmatrix} -10 \\ -8 \end{bmatrix}$

We have, Image = T.M. × object or, $\begin{bmatrix} -10 \\ -8 \end{bmatrix}$ = $\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix}$ × $\begin{bmatrix} a \\ b \end{bmatrix}$
or, $\begin{bmatrix} -10 \\ -8 \end{bmatrix}$ = $\begin{bmatrix} -2b \\ -2a \end{bmatrix}$

Comparing corresponding value of equal matrix,
a = 5, b = 4

Find the transformation matrix which represents the reflection on the line y = x.

Let, Object = $\begin{bmatrix} x \\ y \end{bmatrix}$ and Transformation matrix (T.M.) = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
Image on reflection on y = x is $\begin{bmatrix} y \\ x \end{bmatrix}$

Now, we know, Image = T.M. × Object or, $\begin{bmatrix} y \\ x \end{bmatrix}$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
or, $\begin{bmatrix} y \\ x \end{bmatrix}$ = $\begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$

Above condition will be true if,
a = 0, b = 1
c = 1, d = 0

So, Transformation matrix = $\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$

Find the coordinates of the image of the point A(-4, 3) and B(6, -2) under the transformation 2 × 2 matrix $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$.

Here,
Transformation matrix (T.M.) = $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$
Object = A = $\begin{bmatrix} -4 \\ 3 \end{bmatrix}$ and B = $\begin{bmatrix} 6\\ -2 \end{bmatrix}$
Image = A' and B'

Now, we have, Image = T.M. × Object or, A' = $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ $\begin{bmatrix} -4 \\ 3 \end{bmatrix}$
So, A' = $\begin{bmatrix} 4 \\ -3 \end{bmatrix}$

Again, Image = T.M. × Object or, B' = $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ $\begin{bmatrix} 6\\ -2 \end{bmatrix}$
So, B' = $\begin{bmatrix} -6 \\ 2 \end{bmatrix}$

Hence, A' is (4, -3) and B' is (-6, 2).

What transformation does the matrix $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ denote? Using the given matrix, find the image of the point (-3, 2).

First part,
Given, Transformation matrix (T.M.) = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$
Let, object = $\begin{bmatrix} x \\ y \end{bmatrix}$

Now, we know,
Image = T.M. × Object = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} x \\ y \end{bmatrix}$
= $\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}$
= $\begin{bmatrix} -y \\ -x \end{bmatrix}$

So, the given transformation matrix represents reflection on the line x = -y.

Second part,
Object = A = $\begin{bmatrix} -3 \ 2 \end{bmatrix}$

Now, We know, Image = T.M. × Object or, A' = $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ × $\begin{bmatrix} -3 \\ 2 \end{bmatrix}$
so, A' = $\begin{bmatrix} -2 \\ 3 \end{bmatrix}$