Chapter 11.3, Problem 65AYU

To determine

To find: If row 2 of a 3 by 3 determinant is multiplied by k and the result is added t the entires in row 1 , there is no change in the value of the determinant.

Answer to Problem 65AYU

  $\left|B\right|=0$

Hence we see that there is no change in the value of the determinant

Explanation of Solution

Given information: $A=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

Calculation:

let there be a 3 by 3 determinant (A)

  $A=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

When row 2 is multiplied by k.

  $A=\left|\begin{array}{ccc}1& 1& 1\\ k& k& k\\ 1& 1& 1\end{array}\right|$

The value of determinant can be given as

  $\left|A\right|=1(k-k)-1(k-k)+1(k-k)$

  $\left|A\right|=0$

When the result is added to the determinant (A)

  $B=\left|\begin{array}{ccc}1+0& 1+0& 1+0\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

  $B=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

The value of determinant B is

  $\left|B\right|=1(1-1)-1(1-1)+1(1-1)$

  $\left|B\right|=0$

Hence we see that there is no change in the value of the determinant.