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Matrix Transformation

Which transformation does the matrix [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} represent?

Given, Transformation matrix (T.M.) = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
= [(0)x+(−1)y(−1)x+(0)y]\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}
= [−y−x]\begin{bmatrix} -y \\ -x \end{bmatrix}

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

Which transformation does the matrix [−1001]\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} represent?

Given, Transformation matrix (T.M.) = [−1001]\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [−1001]\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
= [(−1)x+(0)y(0)x+(1)y]\begin{bmatrix} (-1)x + (0)y \\ (0)x + (1)y \end{bmatrix}
= [−xy]\begin{bmatrix} -x \\ y \end{bmatrix}

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

Which transformation does the matrix [0−1−10]\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.) = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
= [(0)x+(−1)y(−1)x+(0)y]\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}
= [−y−x]\begin{bmatrix} -y \\ -x \end{bmatrix}

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

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

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

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

Now, we know, Image = T.M. × Object or, [−y−x]\begin{bmatrix} -y \\ -x \end{bmatrix} = [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
or, [−y−x]\begin{bmatrix} -y \\ -x \end{bmatrix} = [ax+bycx+dy]\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 = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}

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

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

Now, we know, Image = T.M. × Object or, [−xy]\begin{bmatrix} -x \\ y \end{bmatrix} = [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
or, [−xy]\begin{bmatrix} -x \\ y \end{bmatrix} = [ax+bycx+dy]\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 = [−1001]\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}

What transformation does the matrix [1001]\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.) = [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [1001]\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} × [xy]\begin{bmatrix} x \ y \end{bmatrix}
= [(1)x+(0)y(0)x+(1)y]\begin{bmatrix} (1)x + (0)y \ (0)x + (1)y \end{bmatrix}
= [xy]\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 (pq)\begin{pmatrix} p \\ q \end{pmatrix}
So, (a, b) →T(pq)\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 = (2−3)\begin{pmatrix} 2 \\ -3 \end{pmatrix}

Now, using 2 ×1 matrix to transform the point (5, 7),
(5, 7) →T(2−3)\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 (pq)\begin{pmatrix} p \\ q \end{pmatrix}
So, (a, b) →T(pq)\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 = (4−5)\begin{pmatrix} 4 \\ -5 \end{pmatrix}

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

What transformation does the matrix [0−1−10]\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.) = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
= [(0)x+(−1)y(−1)x+(0)y]\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}
= [−y−x]\begin{bmatrix} -y \\ -x \end{bmatrix}

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

Second part,
Object = A = [5−7]\begin{bmatrix} 5 \\ -7 \end{bmatrix}

Now, We know, Image = T.M. × Object or, A' = [0−1−10]\begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} × [5−7]\begin{bmatrix} 5 \ -7 \end{bmatrix}
so, A' = [7−5]\begin{bmatrix} 7 \ -5 \end{bmatrix}

What transformation does the matrix [01−10]\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.) = [01−10]\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [01−10]\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
= [(0)x+(1)y(−1)x+(0)y]\begin{bmatrix} (0)x + (1)y \\ (-1)x + (0)y \end{bmatrix}
= [y−x]\begin{bmatrix} y \\ -x \end{bmatrix}

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

Second part,
Object = A = [−45]\begin{bmatrix} -4 \\ 5 \end{bmatrix}

Now, We know, Image = T.M. × Object or, A' = [01−10]\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} × [−45]\begin{bmatrix} -4 \ 5 \end{bmatrix}
so, A' = [54]\begin{bmatrix} 5 \ 4 \end{bmatrix}

If P(a, b) is transformed by [0−2−20]\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix} to P'(-10, -8), find the value of a and b.

Here, Object = P = [ab]\begin{bmatrix} a \\ b \end{bmatrix}
Transformation matrix (T.M.) = [0−2−20]\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix}
Image = P' = [−10−8]\begin{bmatrix} -10 \\ -8 \end{bmatrix}

We have, Image = T.M. × object or, [−10−8]\begin{bmatrix} -10 \\ -8 \end{bmatrix} = [0−2−20]\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix} × [ab]\begin{bmatrix} a \\ b \end{bmatrix}
or, [−10−8]\begin{bmatrix} -10 \\ -8 \end{bmatrix} = [−2b−2a]\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 = [xy]\begin{bmatrix} x \\ y \end{bmatrix} and Transformation matrix (T.M.) = [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix}
Image on reflection on y = x is [yx]\begin{bmatrix} y \\ x \end{bmatrix}

Now, we know, Image = T.M. × Object or, [yx]\begin{bmatrix} y \\ x \end{bmatrix} = [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
or, [yx]\begin{bmatrix} y \\ x \end{bmatrix} = [ax+bycx+dy]\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 = [0110]\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 [−100−1]\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}.

Here,
Transformation matrix (T.M.) = [−100−1]\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
Object = A = [−43]\begin{bmatrix} -4 \\ 3 \end{bmatrix} and B = [6−2]\begin{bmatrix} 6\\ -2 \end{bmatrix}
Image = A' and B'

Now, we have, Image = T.M. × Object or, A' = [−100−1]\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} [−43]\begin{bmatrix} -4 \\ 3 \end{bmatrix}
So, A' = [4−3]\begin{bmatrix} 4 \\ -3 \end{bmatrix}

Again, Image = T.M. × Object or, B' = [−100−1]\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} [6−2]\begin{bmatrix} 6\\ -2 \end{bmatrix}
So, B' = [−62]\begin{bmatrix} -6 \\ 2 \end{bmatrix}

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

What transformation does the matrix [0−1−10]\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.) = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
Let, object = [xy]\begin{bmatrix} x \\ y \end{bmatrix}

Now, we know,
Image = T.M. × Object = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} × [xy]\begin{bmatrix} x \\ y \end{bmatrix}
= [(0)x+(−1)y(−1)x+(0)y]\begin{bmatrix} (0)x + (-1)y \\ (-1)x + (0)y \end{bmatrix}
= [−y−x]\begin{bmatrix} -y \\ -x \end{bmatrix}

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

Second part,
Object = A = [−32]\begin{bmatrix} -3 \ 2 \end{bmatrix}

Now, We know, Image = T.M. × Object or, A' = [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} × [−32]\begin{bmatrix} -3 \\ 2 \end{bmatrix}
so, A' = [−23]\begin{bmatrix} -2 \\ 3 \end{bmatrix}

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