Exploration

Evaluate the determinant of each.

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

Evaluate the determinant of each.

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

Verify that the determinant of an upper-triangular $3\!\times\!3$ matrix is the product down the diagonal.

Do lower-triangular matrices work the same way?

Use the determinant to decide if each is singular or nonsingular.

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

Singular or nonsingular? Use the determinant to decide.

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

Each pair of matrices differ by one row operation. Use this operation to compare $\det(A)$ with $\det(B)$.

• $A=\begin{pmatrix}1&2\\ 2&3\end{pmatrix}$, $B=\begin{pmatrix}1&2\\ 0&-1\end{pmatrix}$

• $A=\begin{pmatrix}3&1&0\\ 0&0&1\\ 0&1&2\end{pmatrix}$, $B=\begin{pmatrix}3&1&0\\ 0&1&2\\ 0&0&1\end{pmatrix}$

• $A=\begin{pmatrix}1&-1&3\\ 2&2&-6\\ 1&0&4\end{pmatrix}$, $B=\begin{pmatrix}1&-1&3\\ 1&1&-3\\ 1&0&4\end{pmatrix}$

Find the determinant of this $4\!\times\!4$ matrix by following the plan: perform Gauss’s Method and look for the determinant to remain unchanged on a row combination, to change sign on a row swap, to rescale on the rescaling of a row, and such that the determinant of the echelon form matrix is the product down its diagonal.

Show this.

Which real numbers $x$ make this matrix singular?

Do the Gaussian reduction to check the formula for $3\!\times\!3$ matrices stated in the preamble to this section.

$\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}$ is nonsingular iff $aei+bfg+cdh-hfa-idb-gec\neq 0$

Show that the equation of a line in $\mathbb{R}^{2}$ through $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by this determinant.

Many people have learned this mnemonic for the determinant of a $3\!\times\!3$ matrix: copy the first two columns to the right side of the matrix, then take the products down the forward diagonals and add them together, and then take the products on the backward diagonals and subtract them. That is, first write

and then calculate this.

• Check that this agrees with the formula given in the preamble to this s…

The cross product of the vectors

is the vector computed as this determinant.

Note that the first row’s entries are vectors, the vectors from the standard basis for $\mathbb{R}^{3}$. Show that the cross product of two vectors is perpendicular to each vector.

Prove that each statement holds for $2\!\times\!2$ matrices.

• The determinant of a product is the product of the determinants $\det(ST)=\det(S)\cdot\det(T)$.

• If $T$ is invertible then the determinant of the inverse is the inverse of the determinant $\det(T^{-1})=(\,\det(T)\,)^{-1}$.

Matrices $T$ and $T^{\prime}$ are similar if there is a nonsingular matrix $P$ such that $T^{\prime}=PTP^{-1}$. (We shall look at this relationship in Chapter Five.) Show that similar $2\!\times\!2$ matrices have the same determinant.

Prove that for $2\!\times\!2$ matrices, the determinant of a matrix equals the determinant of its transpose. Does that also hold for $3\!\times\!3$ matrices?

Is the determinant function linear  — is $\det(x\cdot T+y\cdot S)=x\cdot\det(T)+y\cdot\det(S)$?

Show that if $A$ is $3\!\times\!3$ then $\det(c\cdot A)=c^{3}\cdot\det(A)$ for any scalar $c$.

Which real numbers $\theta$ make

singular? Explain geometrically.

If a third order determinant has elements $1$, $2$, …, $9$, what is the maximum value it may have?