Chapter 2, Problem 15RP

Explanation of Solution

a.

Existence of Inverse:

Consider the following matrix,

$A=[\begin{array}{l}\text{a 0 0 0}\\ \text{0 b 0 0}\\ \text{0 0 c 0}\\ \text{0 0 0 \&\#}\end{array}$

Explanation of Solution

b.

Finding the inverse of the given matrix:

Consider the following matrix,

$A=\left[\begin{array}{l}\text{a 0 0 0}\\ \text{0 b 0 0}\\ \text{0 0 c 0}\\ \text{0 0 0 d}\end{array}\right]$

Suppose the inverse of the matrix $A$ is as follows:

$A^{-1}=\left[\begin{array}{l}\text{e 0 0 0}\\ \text{0 f 0 0}\\ \text{0 0 g 0}\\ \text{0 0 0 h}\end{array}\right]$

Then we have to solve the following equation

$\left[\begin{array}{l}\text{a 0 0 0}\\ \text{0 b 0 0}\\ \text{0 0 c 0}\\ \text{0 0 0 d}\end{array}\right]\left[\begin{array}{l}e\\ 0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}\right]$

This gives us

$e=\frac{1}{a}$

Also, we have

$\left[\begin{array}{l}\text{a 0 0 0}\\ \text{0 b 0 0}\\ \text{0 0 c 0}\\ \text{0 0 0 d}\end{array}\right]\left[\begin{array}{l}0\\ f\\ 0\\ 0\end{array}\right]=$