Chapter 8, Problem 19PS

To determine

Find the determinant of a matrix which satisfies the given statement.

Answer to Problem 19PS

The determinant of $A$ is an $n\times n$ matrix each of whose rows adds up to zero is zero.

Explanation of Solution

Given:

The given statement is that the $A$ is an $n\times n$ matrix each of whose rows adds up to zero.

Calculation:

Let the three square matrices are

  $2\times 2$ matrix is

  $\begin{array}{l}{M}_1=\left[\begin{array}{cc}-4& 4\\ 5& -5\end{array}\right]\\ D_1=\left|\begin{array}{cc}-4& 4\\ 5& -5\end{array}\right|=0\end{array}$

  $3\times 3$ matrix is

  $\begin{array}{l}{M}_2=\left[\begin{array}{ccc}-3& 2& 1\\ 3& -7& 4\\ -9& 4& 5\end{array}\right]\\ D_2=\left|\begin{array}{ccc}-3& 2& 1\\ 3& -7& 4\\ -9& 4& 5\end{array}\right|=0\end{array}$

  $4\times 4$ matrix is

  $\begin{array}{l}{M}_3=\left[\begin{array}{cccc}-6& 2& 1& 3\\ 4& -10& 2& 4\\ 3& -7& 4& 0\\ -9& 4& 4& 1\end{array}\right]\\ D_3=\left|\begin{array}{cccc}-6& 2& 1& 3\\ 4& -10& 2& 4\\ 3& -7& 4& 0\\ -9& 4& 4& 1\end{array}\right|=0\end{array}$

Hence the determinant of $A$ is an $n\times n$ matrix each of whose rows adds up to zero is zero.