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.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }(AB)=(\text{det }A)(\text{det }B)}
2.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }I=n}
3.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A^T=\text{det }A}

Solution:

Step 1:

Step 2:

Final Answer:

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