Matrix

[fismaquim]

In a class on matrices, the teacher defined the concept of the orthogonal matrix to students and went further by citing its properties, saying that the module of the determinant of an orthogonal matrix is ​​always equal to 1 and also the applications in matrix decomposition and linear transformations of rotation. After the explanation, the teacher gave an example of an orthogonal matrix of order 3. One of the students managed, however, to copy only two of its lines, reproduced below:

Imagem4.gif (2.33 KiB) Viewed 2044 times

At home, remembering that it was an orthogonal matrix, the student could conclude that the third line could be:

a) 2/3, -2/3 e 1/3
b) 1/3, 2/3 e 2/3
c) 2/3, 1/3 e -2/3
d) 1/3, -2/3 e 1/3
e) 2/3, 1/3 e 2/3

[HallsofIvy]

Let the last row be "a b c". then, expanding the determinant on that last row,
[tex]a\left|\begin{array}{cc}-\frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & \frac{2}{3}\end{array}\right|-[/tex][tex]b\left|\begin{array}{cc}\frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3}\end{array}\right|+[/tex][tex]c\left|\begin{array}{cc}\frac{2}{3} & -\frac{2}{3} \\ \frac{1}{3} & \frac{2}{3}\end{array}\right|[/tex]
[tex]= a\left(-\frac{4}{9}- \frac{2}{9}\right)- b\left(\frac{4}{9}- \frac{1}{9}\right)+ c\left(\frac{4}{9}+ \frac{2}{9}\right)[/tex]
[tex]-\frac{2}{3}a- \frac{1}{3}b+ \frac{2}{3}c[/tex].

Since this question is multiple choice you can just try each of those.

[Guest]

This has been here since July so

As I showed before the deteminant must be [tex]-\frac{2}{3}]a- \frac{1}{3}b+ \frac{2}{3}c[/tex] and to be an orthogonal matrix that must be either 1 or -1.

The possible answers are
a= 2/3, b= -2/3, d= 1/3
That would give -2/3(2/3)+ 1/3(-2/3)+ 2/3(1/3)= -4/9- 2/9+ 2/9= -4/9. No, that is neither 1 nor -1.

a= 1/3, b= 2/3, c= 2/3
That would give -2/3(1/3)- 1/3(2/3)+ 2/3(2/3)= -2/9- 2/9+ 4/9= 0,
No, that is neither 1 nor -1,

a= 2/3, b= 1/3, c= -2/3
That would give -2/3(2/3)- 1/3(1/3)+ 2/3(-2/3)= -4/9- 1/9- 4/9= -1,
Aha! This appears to be the one.

Just to be complete:
a= 1/3, b= -2/3, c= 1/3
That would give -2/3(1/3)- 1/3(-2/3)+ 2/3(1/3)= -2/9+ 2/9+ 2/9= 0,
No, that is neither 1 nor -1,

a= 2/3, b= 1/3, c= 2/3
That would give -2/3(2/3)- 1/3(1/3)+ 2/3(2/3)= -4/9- 1/9+ 4/9= 1/9.
No, that is neither 1 nor -1,