# 2.7.1: Exercises 2.7

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In Exercises $$\PageIndex{1}$$ - $$\PageIndex{4}$$, matrices $$A$$ and $$B$$ are given. Compute $$(AB)^{-1}$$ and $$B^{-1}A^{-1}$$.

##### Exercise $$\PageIndex{1}$$

$$A=\left[\begin{array}{cc}{1}&{2}\\{1}&{1}\end{array}\right],\quad B=\left[\begin{array}{cc}{3}&{5}\\{2}&{5}\end{array}\right]$$

$$(AB)^{-1}=\left[\begin{array}{cc}{-2}&{3}\\{1}&{-1.4}\end{array}\right]$$

##### Exercise $$\PageIndex{2}$$

$$A=\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right],\quad B=\left[\begin{array}{cc}{7}&{1}\\{2}&{1}\end{array}\right]$$

$$(AB)^{-1}=\left[\begin{array}{cc}{-7/10}&{3/10}\\{29/10}&{-11/10}\end{array}\right]$$

##### Exercise $$\PageIndex{3}$$

$$A=\left[\begin{array}{cc}{2}&{5}\\{3}&{8}\end{array}\right],\quad B=\left[\begin{array}{cc}{1}&{-1}\\{1}&{4}\end{array}\right]$$

$$(AB)^{-1}=\left[\begin{array}{cc}{29/5}&{-18/5}\\{-11/5}&{7/5}\end{array}\right]$$

##### Exercise $$\PageIndex{4}$$

$$A=\left[\begin{array}{cc}{2}&{4}\\{2}&{5}\end{array}\right],\quad B=\left[\begin{array}{cc}{2}&{2}\\{6}&{5}\end{array}\right]$$

$$(AB)^{-1}=\left[\begin{array}{cc}{-29/4}&{6}\\{17/2}&{-7}\end{array}\right]$$

In Exercises $$\PageIndex{5}$$ - $$\PageIndex{8}$$, a $$2\times 2$$ matrix $$A$$ is given. Compute $$A^{-1}$$ and $$(A^{-1})^{-1}$$ using Theorem 2.6.3.

##### Exercise $$\PageIndex{5}$$

$$A=\left[\begin{array}{cc}{-3}&{5}\\{1}&{-2}\end{array}\right]$$

$$A^{-1}=\left[\begin{array}{cc}{-2}&{-5}\\{-1}&{-3}\end{array}\right],\quad (A^{-1})^{-1}=\left[\begin{array}{cc}{-3}&{5}\\{1}&{-2}\end{array}\right]$$

##### Exercise $$\PageIndex{6}$$

$$A=\left[\begin{array}{cc}{3}&{5}\\{2}&{4}\end{array}\right]$$

$$A^{-1}=\left[\begin{array}{cc}{2}&{-5/2}\\{-1}&{3/2}\end{array}\right],\quad (A^{-1})^{-1}=\left[\begin{array}{cc}{3}&{5}\\{2}&{4}\end{array}\right]$$

##### Exercise $$\PageIndex{7}$$

$$A=\left[\begin{array}{cc}{2}&{7}\\{1}&{3}\end{array}\right]$$

$$A^{-1}=\left[\begin{array}{cc}{-3}&{7}\\{1}&{-2}\end{array}\right],\quad (A^{-1})^{-1}=\left[\begin{array}{cc}{2}&{7}\\{1}&{3}\end{array}\right]$$

##### Exercise $$\PageIndex{8}$$

$$A=\left[\begin{array}{cc}{9}&{0}\\{7}&{9}\end{array}\right]$$

$$A^{-1}=\left[\begin{array}{cc}{1/9}&{0}\\{-7/81}&{1/9}\end{array}\right],\quad (A^{-1})^{-1}=\left[\begin{array}{cc}{9}&{0}\\{7}&{9}\end{array}\right]$$

##### Exercise $$\PageIndex{9}$$

Find $$2\times 2$$ matrices $$A$$ and $$B$$ that are each invertible, but $$A + B$$ is not.

Solutions will vary.

##### Exercise $$\PageIndex{10}$$

Create a random $$6\times 6$$ matrix $$A$$, then have a calculator or computer compute $$AA^{−1}$$. Was the identity matrix returned exactly? Comment on your results.

Likely some entries that should be 0 will not be exactly 0, but rather very small values

##### Exercise $$\PageIndex{11}$$

Use a calculator or computer to compute $$AA^{-1}$$, where

$A=\left[\begin{array}{cccc}{1}&{2}&{3}&{4}\\{1}&{4}&{9}&{16}\\{1}&{8}&{27}&{64}\\{1}&{16}&{81}&{256}\end{array}\right]. \nonumber$

Was the identity matrix returned exactly. Comment on your results.