Chapter 10.6, Problem 66E

a.

To determine

The matrix with a row or column consisting entirely of zeros.

Answer to Problem 66E

The matrix with a row or column consisting entirely of zeros is $\left|A\right|=0$

Explanation of Solution

Calculation:

  $\begin{array}{l}A=\left[\begin{array}{ccc}a& d& 0\\ b& e& 0\\ c& f& 0\end{array}\right]\\ \left|A\right|=0\end{array}$

Conclusion:

Hence, the matrix with a row or column consisting entirely of zeros is $\left|A\right|=0$

b.

To determine

A matrix with two rows the same or two columns the same.

Answer to Problem 66E

Proved that the matrix with two rows the same or two columns the same.

Explanation of Solution

Calculation:

  $\begin{array}{l}A=\left[\begin{array}{ccc}a& a& d\\ b& b& e\\ c& c& f\end{array}\right]\\ \left|A\right|=a\left(bf-ce\right)-a\left(bf-ce\right)-a\left(bc-bc\right)\\ \left|A\right|=0\end{array}$

Conclusion:

Hence, matrix with two rows the same or two columns the same is proved.

c.

To determine

A matrix in which one row is a multiple of another row, or one column is a multiple of another column.

Answer to Problem 66E

The matrix in which one row is a multiple of another row, or one column is a multiple of another column is $\left|A\right|=0$

.

Explanation of Solution

Calculation:

  $\begin{array}{l}A=\left[\begin{array}{ccc}a& \lambda a& d\\ b& \lambda b& e\\ c& \lambda c& f\end{array}\right]\\ \left|A\right|=\lambda \left[\begin{array}{ccc}a& a& d\\ b& b& e\\ c& c& f\end{array}\right]\\ \left|A\right|=\lambda \left(0\right)\\ \left|A\right|=0\end{array}$

Conclusion:

Hence, the matrix in which one row is a multiple of another row, or one column is a multiple of another column is $\left|A\right|=0$ .