Properties of Determinants

Find each determinant by performing one row operation.

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

Use Gauss’s Method to find each determinant.

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

Use Gauss’s Method to find each.

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

For which values of $k$ does this system have a unique solution?

Express each of these in terms of $\mathchoice{\left|H\right|}{|H|}{|H|}{|H|}$.

• $\begin{vmatrix}h_{3,1}&h_{3,2}&h_{3,3}\\ h_{2,1}&h_{2,2}&h_{2,3}\\ h_{1,1}&h_{1,2}&h_{1,3}\end{vmatrix}$

• $\begin{vmatrix}-h_{1,1}&-h_{1,2}&-h_{1,3}\\ -2h_{2,1}&-2h_{2,2}&-2h_{2,3}\\ -3h_{3,1}&-3h_{3,2}&-3h_{3,3}\end{vmatrix}$

• $\begin{vmatrix}h_{1,1}+h_{3,1}&h_{1,2}+h_{3,2}&h_{1,3}+h_{3,3}\\ h_{2,1}&h_{2,2}&h_{2,3}\\ 5h_{3,1}&5h_{3,2}&5h_{3,3}\end{vmatrix}$

Find the determinant of a diagonal matrix.

Describe the solution set of a homogeneous linear system if the determinant of the matrix of coefficients is nonzero.

Show that this determinant is zero.

• Find the $1\!\times\!1$, $2\!\times\!2$, and $3\!\times\!3$ matrices with $i,j$ entry given by $(-1)^{i+j}$.

• Find the determinant of the square matrix with $i,j$ entry $(-1)^{i+j}$.

• Find the $1\!\times\!1$, $2\!\times\!2$, and $3\!\times\!3$ matrices with $i,j$ entry given by $i+j$.

• Find the determinant of the square matrix with $i,j$ entry $i+j$.

Show that determinant functions are not linear by giving a case where $\mathchoice{\left|A+B\right|}{|A+B|}{|A+B|}{|A+B|}\neq\mathchoice{\left|A\right|}{|A|}{|A|}{|A|}+\mathchoice{\left|B\right|}{|B|}{|B|}{|B|}$.

The second condition in the definition, that row swaps change the sign of a determinant, is somewhat annoying. It means we have to keep track of the number of swaps, to compute how the sign alternates. Can we get rid of it? Can we replace it with the condition that row swaps leave the determinant unchanged? (If so then we would need new $1\!\times\!1$, $2\!\times\!2$, and $3\!\times\!3$ formulas, but that would be a minor matter.)

Prove that the determinant of any triangular matrix, upper or lower, is the product down its diagonal.

Refer to the definition of elementary matrices in the Mechanics of Matrix Multiplication subsection.

• What is the determinant of each kind of elementary matrix?

• Prove that if $E$ is any elementary matrix then $\mathchoice{\left|ES\right|}{|ES|}{|ES|}{|ES|}=\mathchoice{\left|E\right|}{|E|}{|E|}{|E|}\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$ for any appropriately sized $S$.

• (This question doesn’t involve determinants.) Prove that if $T$ is singular then a product $TS$ is also singular.

• Show that $\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}=\mathchoice{\left|T\right|}{|T|}{|T|}{|T|}\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$.

• Show that if $T$ is nonsingular…

Prove that the determinant of a product is the product of the determinants $\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}=\mathchoice{\left|T\right|}{|T|}{|T|}{|T|}\,\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$ in this way. Fix the $n\!\times\!n$ matrix $S$ and consider the function $d\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}$ given by $T\mapsto\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}/\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$.

• Check that $d$ satisfies condition (1) in the definition of a determinant function.

• Check condition (2).

• Check condition (3).

• Check condition (4).

• Conclude the determinant of a product is the product of the determinants.

A submatrix of a given matrix $A$ is one that we get by deleting some of the rows and columns of $A$. Thus, the first matrix here is a submatrix of the second.

Prove that for any square matrix, the rank of the matrix is $r$ if and only if $r$ is the largest integer such that there is an $r\!\times\!r$ submatrix with a nonzero determinant.

Prove that a matrix with rational entries has a rational determinant.

Find the element of likeness in (a) simplifying a fraction, (b) powdering the nose, (c) building new steps on the church, (d) keeping emeritus professors on campus, (e) putting $B$, $C$, $D$ in the determinant