Inverses

Supply the intermediate steps in Example 4.9.

Use Corollary 4.11 to decide if each matrix has an inverse.

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

For each invertible matrix in the prior problem, use Corollary 4.11 to find its inverse.

Find the inverse, if it exists, by using the Gauss-Jordan Method. Check the answers for the $2\!\times\!2$ matrices with Corollary 4.11.

• (a)  $\begin{pmatrix}3&1\\ 0&2\end{pmatrix}$     (b)  $\begin{pmatrix}2&1/2\\ 3&1\end{pmatrix}$     (c)  $\begin{pmatrix}2&-4\\ -1&2\end{pmatrix}$     (d)  $\begin{pmatrix}1&1&3\\ 0&2&4\\ -1&1&0\end{pmatrix}$     (e)  $\begin{pmatrix}0&1&5\\ 0&-2&4\\ 2&3&-2\end{pmatrix}$     (f)  $\begin{pmatrix}2&2&3\\ 1&-2&-3\\ 4&-2&-3\end{pmatrix}$

What matrix has this one for its inverse?

How does the inverse operation interact with scalar multiplication and addition of matrices?

• What is the inverse of $rH$?

• Is $(H+G)^{-1}=H^{-1}+G^{-1}$?

Is $(T^{k})^{-1}=(T^{-1})^{k}$?

Is $H^{-1}$ invertible?

For each real number $\theta$ let $t_{\theta}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ be represented with respect to the standard bases by this matrix.

Show that $t_{\theta_{1}+\theta_{2}}=t_{\theta_{1}}\cdot t_{\theta_{2}}$. Show also that ${t_{\theta}}^{-1}=t_{-\theta}$.

Do the calculations for the proof of Corollary 4.11.

Show that this matrix

has infinitely many right inverses. Show also that it has no left inverse.

In the review of inverses example, starting this subsection, how many left inverses has $\iota$?

If a matrix has infinitely many right-inverses, can it have infinitely many left-inverses? Must it have?

Assume that $g\colon V\to W$ is linear. One of these is true, the other is false. Which is which?

• If $f\colon W\to V$ is a left inverse of $g$ then $f$ must be linear.

• If $f\colon W\to V$ is a right inverse of $g$ then $f$ must be linear.

Assume that $H$ is invertible and that $HG$ is the zero matrix. Show that $G$ is a zero matrix.

Prove that if $H$ is invertible then the inverse commutes with a matrix $GH^{-1}=H^{-1}G$ if and only if $H$ itself commutes with that matrix $GH=HG$.

Show that if $T$ is square and if $T^{4}$ is the zero matrix then $(I-T)^{-1}=I+T+T^{2}+T^{3}$. Generalize.

Let $D$ be diagonal. Describe $D^{2}$, $D^{3}$, … , etc. Describe $D^{-1}$, $D^{-2}$, … , etc. Define $D^{0}$ appropriately.

Prove that any matrix row-equivalent to an invertible matrix is also invertible.

The first question below appeared as Exercise 17.

• Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each.

• Show that if $T$ and $S$ are square then $TS=I$ if and only if $ST=I$.

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

• Show that ${{(GH)}^{\mathsf{T}}}={{H}^{\mathsf{T}}}{{G}^{\mathsf{T}}}$.

• A square matrix is symmetric if each $i,j$ entry equals the $j,i$ entry (that is, if the matrix equals its transpose). Show that the matrices $H{{H}^{\mathsf{T}}}$ and ${{H}^{\mathsf{T}}}H$ are symmetric.

• Show that the inverse of the transpose is the transpose of the inverse.

• Show that the inverse of a symmetric matrix is symmetric.

• Prove that the composition of the projections $\pi_{x},\pi_{y}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ is the zero map despite that neither is the zero map.

• Prove that the composition of the derivatives $d^{2}/dx^{2},\,d^{3}/dx^{3}\colon\mathcal{P}_{4}\to\mathcal{P}_{4}$ is the zero map despite that neither map is the zero map.

• Give matrix equations representing each of the prior two items.

When two things multiply to give zero despite that neither is zero, each is said to be a zero divisor. Prove that no zer…

In the algebra of real numbers, quadratic equations have at most two solutions. Matrix algebra is different. Show that the $2\!\times\!2$ matrix equation $T^{2}=I$ has more than two solutions.

Is the relation ‘is a two-sided inverse of’ transitive? Reflexive? Symmetric?

Prove: if the sum of the elements of each row of a square matrix is $k$, then the sum of the elements in each row of the inverse matrix is $1/k$.