Chapter 2, Problem 21RP

Explanation of Solution

Infinite number of solutions:

Consider the following linear system,

$\left[x_1{x}_2\dots x_n\right]=\left[x_1{x}_2\dots x_n\right]P$

Where $P$ is as follows:

$P=\left[\begin{array}{l}{p}_{11}{p}_{12}\cdots p_{1n}\\ p_{21}{p}_{22}\cdots p_{2n}\\ ⋮⋮\ddots ⋮\\ p_{n1}{p}_{n2}\cdots p_{nn}\end{array}\right]$

This system can be written as given below:

$\left[x_1{x}_2\dots x_n\right]\left[P-I\right]=0$

Therefore, we have,

$\left[P-I\right]=\left[\begin{array}{l}{p}_{11}\text{-1 }{p}_{12}\cdots p_{1n}\\ p_{21}{p}_{22}\text{-1 }\cdots p_{2n}\\ ⋮⋮\ddots ⋮\\ p_{n1}{p}_{n2}\cdots p_{nn-1}\end{array}\right]$

We know that the transpose of the matrix does not affect the rank of the matrix.

Therefore, we get

${\left[P-I\right]}^T=\left[\begin{array}{l}{p}_{11}\text{-1 }{p}_{21}\cdots p_{n1}\\ p_{12}{p}_{22}\text{-1 }\cdots p_{n2}\\ ⋮⋮\ddots ⋮\\ p_{1n}{p}_{2n}\cdots p_{nn-1}\end{array}\right]$

Adding all the rows in the last row will give

${[P\&}^{}$