031 Review Part 2, Problem 9

If  $A$  is an  $n\times n$  matrix such that  $AA^{T}=I,$  what are the possible values of  ${\text{det }}A?$

Foundations:
Recall:
1.  ${\text{det }}(AB)=({\text{det }}A)({\text{det }}B)$
2.  ${\text{det }}I=n$
3.  ${\text{det }}A^{T}={\text{det }}A$

Solution:

Step 1:
Using the facts in the Foundations section, we have
${\begin{array}{rcl}\displaystyle {1}&=&\displaystyle {{\text{det }}(I)}\\&&\\&=&\displaystyle {{\text{det }}(AA^{T})}\\&&\\&=&\displaystyle {({\text{det }}A)({\text{det }}A^{T})}\\&&\\&=&\displaystyle {({\text{det }}A)({\text{det }}A)}\\&&\\&=&\displaystyle {({\text{det }}A)^{2}.}\end{array}}$

Step 2:
Taking the square root of both sides of the equation
$({\text{det }}A)^{2}=1,$
we obtain  ${\text{det }}A=\pm 1.$

Final Answer:
${\text{det }}A=\pm 1$

Return to Sample Exam