Chapter 11.3, Problem 6AYU

To determine

To explain: Whether the statement “If any row (or any column) of a determinant is multiplied by a nonzero number k , the value of determinant remains unchanged.” is true or false.

Answer to Problem 6AYU

The statement “If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of determinant remains unchanged.” is false.

Explanation of Solution

Given information:

The statement “If any row (or any column) of a determinant is multiplied by a nonzero number k , the value of determinant remains unchanged.”

Consider the provided statement “If any row (or any column) of a determinant is multiplied by a nonzero number k , the value of determinant remains unchanged.”

Consider the matrix,

  $\begin{array}{cc}4u& -1\\ -1& 2v\end{array}$

Recall that for the matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

Therefore, determinant of the matrix is given by,

  $\begin{array}{c}\begin{array}{cc}4u& -1\\ -1& 2v\end{array}=\left(4u\right)\left(2v\right)-\left(-1\right)\left(-1\right)\\ =8uv-1\end{array}$

Now, multiply the first row by 2, so new matrix is $\begin{array}{cc}8u& -2\\ -1& 2v\end{array}$ .

Therefore, determinant of the matrix is given by,

  $\begin{array}{c}\begin{array}{cc}8u& -2\\ -1& 2v\end{array}=\left(8u\right)\left(2v\right)-\left(-2\right)\left(-1\right)\\ =16uv-2\\ =2\left(8uv-1\right)\end{array}$

Observe that determinant of new matrix is two times the old one.

Therefore, if any row (or any column) of a determinant is multiplied by a nonzero number k , the value of determinant becomes k times.

Thus, the statement “If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of determinant remains unchanged.” is false.