The Permutation Expansion

For this matrix, find the term associated with each $3$-permutation.

That is, fill in the rest of this table.

For each $3$-permutation $\phi$ find $|P_{\phi}|$.

This determinant is $7$ by the $2\!\times\!2$ formula. Compute it with the permutation expansion.

This determinant is $0$ because the first two rows add to the third. Compute the determinant using the permutation expansion.

Compute the determinant by using the permutation expansion.

• (a)  $\begin{vmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{vmatrix}$     (b)  $\begin{vmatrix}2&2&1\\ 3&-1&0\\ -2&0&5\end{vmatrix}$

Compute these both with Gauss’s Method and the permutation expansion formula.

• (a)  $\begin{vmatrix}2&1\\ 3&1\end{vmatrix}$     (b)  $\begin{vmatrix}0&1&4\\ 0&2&3\\ 1&5&1\end{vmatrix}$

Use the permutation expansion formula to derive the formula for $3\!\times\!3$ determinants.

List all of the $4$-permutations.

A permutation, regarded as a function from the set $\{\hskip 1.00006pt1,..,n\hskip 1.00006pt\}$ to itself, is one-to-one and onto. Therefore, each permutation has an inverse.

• Find the inverse of each $2$-permutation.

• Find the inverse of each $3$-permutation.

Prove that $f$ is multilinear if and only if for all $\vec{v},\vec{w}\in V$ and $k_{1},k_{2}\in\mathbb{R}$, this holds.

How would determinants change if we changed property (4) of the definition to read that $\mathchoice{\left|I\right|}{|I|}{|I|}{|I|}=2$?

Verify the second and third statements in Corollary 3.16.

Show that if an $n\!\times\!n$ matrix has a nonzero determinant then we can express any column vector $\vec{v}\in\mathbb{R}^{n}$ as a linear combination of the columns of the matrix.

True or false: a matrix whose entries are only zeros or ones has a determinant equal to zero, one, or negative one.

• Show that there are $120$ terms in the permutation expansion formula of a $5\!\times\!5$ matrix.

• How many are sure to be zero if the $1,2$ entry is zero?

How many $n$-permutations are there?

Show that the inverse of a permutation matrix is its transpose.

A matrix $A$ is skew-symmetric if ${{A}^{\mathsf{T}}}=-A$, as in this matrix.

Show that $n\!\times\!n$ skew-symmetric matrices with nonzero determinants exist only for even $n$.

What is the smallest number of zeros, and the placement of those zeros, needed to ensure that a $4\!\times\!4$ matrix has a determinant of zero?

If we have $n$ data points $(x_{1},y_{1}),(x_{2},y_{2}),\dots\,,(x_{n},y_{n})$ and want to find a polynomial $p(x)=a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_{1}x+a_{0}$ passing through those points then we can plug in the points to get an $n$ equation/$n$ unknown linear system. The matrix of coefficients for that system is the Vandermonde matrix. Prove that the determinant of the transpose of that matrix of coefficients

equals the product, over all indices $i,j\in\text{$\{\hskip 1.00006pt1,\dots,n\hskip 1.00006pt\}$}$ with $i<j$, of terms of the form $x_{j}-x_{i}$. (This shows that t…

Prove that for any $n\!\times\!n$ matrix $T$ there are at most $n$ distinct reals $r$ such that the matrix $T-rI$ has determinant zero (we shall use this result in Chapter Five).

The nine positive digits can be arranged into $3\!\times\!3$ arrays in $9!$ ways. Find the sum of the determinants of these arrays.

Show that

Let $S$ be the sum of the integer elements of a magic square of order three and let $D$ be the value of the square considered as a determinant. Show that $D/S$ is an integer.

Show that the determinant of the $n^{2}$ elements in the upper left corner of the Pascal triangle

has the value unity.