# Thread: Problem with simple linear algebra exercise

1. ## Problem with simple linear algebra exercise

Good day to all. I apologize if the question asked appears to be a waste of time but it has created some problems for me. In one of my end of chapter exercises I am asked the following:

Find all 2x2 matrices where AA^T (transpose of A) = I

I proceeded with the general equation of A such that

A= [[a,b],[c,d]] where [a,b],[c,d] are the rows of A
A^T = [[a,c],[b,d]]

After AA^T was computed I retrieved the following three equations:

(1) a^2 + b^2 = 1
(2) ac + bd = 0
(3) c^2 + d^2 =1

My first question is this a good method to follow in answering the question? My second question is that I am not sure what conclusions to draw from here (aside that AA^T is symmetric). The other problem is that we have not explicitly covered orthogonal matrices or inverses which are topics I encountered when reading other textbooks about this matter. Any help would be greatly appreciated.

Kindest regards

2. Think about eqn 1. This is satisfied by all the points on a circle of radius 1. You can think of (a,b) as a 2 component vector of magnitude 1 pointing to some point on the circle.

You can reach the same conclusion about (c,d) from eqn.3

Eqn 2 tells you that (a,b) is perpendicular to (c,d) because the dot product is zero.

The net effect is that all solutions look like the hour hand and minute hand at 3:00, but with some arbitrary rotation.

3. Originally Posted by gate13
Good day to all. I apologize if the question asked appears to be a waste of time but it has created some problems for me. In one of my end of chapter exercises I am asked the following:

Find all 2x2 matrices where AA^T (transpose of A) = I

I proceeded with the general equation of A such that

A= [[a,b],[c,d]] where [a,b],[c,d] are the rows of A
A^T = [[a,c],[b,d]]

After AA^T was computed I retrieved the following three equations:

(1) a^2 + b^2 = 1
(2) ac + bd = 0
(3) c^2 + d^2 =1

My first question is this a good method to follow in answering the question? My second question is that I am not sure what conclusions to draw from here (aside that AA^T is symmetric). The other problem is that we have not explicitly covered orthogonal matrices or inverses which are topics I encountered when reading other textbooks about this matter. Any help would be greatly appreciated.

Kindest regards

I think it is an excellent method to answer your question, taking into account that you haven't yet studied orthogonal matrices. Now:

$(2)\,\,ac+bd=0\Longrightarrow a=-\frac{bd}{c}\Longrightarrow\,(1)\,\,1=a^2+b^2=b^2\ left(\frac{d^2}{c^2}+1\right)\Longrightarrow b^2=\frac{c^2}{c^2+d^2}$ $\Longrightarrow b=\pm \frac{c}{\sqrt{c^2+d^2}}\,,\,\,a=\pm \frac{d}{\sqrt{c^2+d^2}}$ , and a similar expression for $b,c$ using $a,b$ in reversed roles.

It's clear from (1) and (3) that $-1\leq a,b,c,d\leq 1$ , and taking a straight-angle triangle with cathetus $c,d$ and hypothenuse $\sqrt{c^2+d^2}$ , we get that we can take

$a=\cos \theta\,,\,\,b=\sin \theta\,,\,\,c=-\sin \theta\,,\,\,d=\cos \theta$ , which gives you the general form of a 2 x 2 orthogonal real matrix.

Tonio

4. Thank you both for your prompt reply to my question. Also, thanks for the geometric interpretation of the problem. Very much appreciated.