Chapter 8, Problem 3PS

a.

To determine

Check whether the given matrix is idempotent or not.

Answer to Problem 3PS

The given matrix is idempotent.

Explanation of Solution

Given:

The given matrix is

  $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

Calculation:

A square matrix is idempotent when $A^2=A$ .

Let $A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

  $\begin{array}{l}{A}^2=A\cdot A\\ =\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\\ =\left[\begin{array}{cc}1\cdot 1+0\cdot 0& 1\cdot 0+0\cdot 0\\ 0\cdot 1+0\cdot 0& 0\cdot 0+0\cdot 0\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\\ =A\end{array}$

Hence the given matrix is idempotent.

b.

To determine

Check whether the given matrix is idempotent or not.

Answer to Problem 3PS

The given matrix is not idempotent.

Explanation of Solution

Given:

The given matrix is

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Calculation:

A square matrix is idempotent when $A^2=A$ .

Let $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

  $\begin{array}{l}{A}^2=A\cdot A\\ =\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\\ =\left[\begin{array}{cc}0\cdot 1+1\cdot 1& 0\cdot 1+1\cdot 0\\ 1\cdot 0+0\cdot 1& 1\cdot 1+0\cdot 0\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ \ne A\end{array}$

Hence the given matrix is not idempotent.

c.

To determine

Check whether the given matrix is idempotent or not.

Answer to Problem 3PS

The given matrix is not idempotent.

Explanation of Solution

Given:

The given matrix is

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

Calculation:

A square matrix is idempotent when $A^2=A$ .

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

  $\begin{array}{l}{A}^2=A\cdot A\\ =\left[\begin{array}{cc}2& 3\\ -1& -2\end{array}\right]\left[\begin{array}{cc}2& 3\\ -1& -2\end{array}\right]\\ =\left[\begin{array}{cc}2\cdot 2+3\cdot \left(-1\right)& 2\cdot 3+3\cdot \left(-2\right)\\ -1\cdot 2+\left(-2\right)\cdot \left(-1\right)& -1\cdot 3+\left(-2\right)\cdot \left(-2\right)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ \ne A\end{array}$

Hence the given matrix is not idempotent.

d.

To determine

Check whether the given matrix is idempotent or not.

Answer to Problem 3PS

The given matrix is not idempotent.

Explanation of Solution

Given:

The given matrix is

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

Calculation:

A square matrix is idempotent when $A^2=A$ .

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

  $\begin{array}{l}{A}^2=A\cdot A\\ =\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]\\ =\left[\begin{array}{cc}2\cdot 2+3\cdot 1& 2\cdot 3+3\cdot 2\\ 1\cdot 2+2\cdot 1& 1\cdot 3+2\cdot 2\end{array}\right]\\ =\left[\begin{array}{cc}7& 12\\ 4& 7\end{array}\right]\\ \ne A\end{array}$

Hence the given matrix is not idempotent.