# 8.5: Review Problems

- Page ID
- 2019

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

1. Find the determinant via expanding by minors.

$$\left(\begin{array}{rrrr} 2 & 1 & 3 & 7 \\ 6 & 1 & 4 & 4 \\ 2 & 1 & 8 & 0 \\ 1 & 0 & 2 & 0\end{array}\right)$$

2. Even if \(M\) is not a square matrix, both \(MM^{T}\) and \(M^{T} M\) are square. Is it true that \(\det(MM^{T}) = \det(M{T} M)\) for all matrices \(M\)? How about \(tr(MM^{T}) = tr(M^{T} M)\)?

3. Let \(\sigma^{-1}\) denote the inverse permutation of \(\sigma\). Suppose the function \(f : {1, 2, 3, 4} \rightarrow R\). Write out explicitly the following two sums:

$$ \sum_{\sigma}f(\sigma(s))~and~\sum_{\sigma}f(\sigma^{-1}(s)).$$

What do you observe? Now write a brief explanation why the following equality holds

$$ \sum_{\sigma}F(\sigma) = \sum_{\sigma}F(\sigma^{-1}).$$

where the domain of the function F is the set of all permutations of n objects.

4. Suppose \(M = LU\) is an \(LU\) decomposition. Explain how you would efficiently compute \(\det M\) in this case. How does this decomposition allow you to easily see if \(M\) is invertible?

5. In computer science, the complexity of an algorithm is (roughly) computed by counting the number of times a given operation is performed. Suppose adding or subtracting any two numbers takes a seconds, and multiplying two numbers takes \(m\) seconds. Then, for example, computing 2·6 -- 5 would take \(a+m\) seconds.

(a) How many additions and multiplications does it take to compute the determinant of a general \(2 \rightarrow 2\) matrix?

(b) Write a formula for the number of additions and multiplications it takes to compute the determinant of a general \(n \rightarrow n\) matrix using the definition of the determinant as a sum over permutations. Assume that finding and multiplying by the sign of a permutation is free.

(c) How many additions and multiplications does it take to compute the determinant of a general \(3 \rightarrow 3\) matrix using expansion by minors? Assuming \(m = 2a\), is this faster than computing the determinant from the definition?

### Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)