Chapter 16.8, Problem 15WE

To determine

To prove: The value of new second order determinant is same as the original one if all entries of either a row or a column are multiplied by a real number k , and the resultant products are added to corresponding entries of either another row or another column.

Explanation of Solution

Given information:

The second order determinant.

Proof:

Consider a second order matrix,

  $\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)$

The determinant of above matrix is,

  $\begin{array}{c}D=\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|\\ =ad-bc\end{array}$

According to the question multiply all entries of row 2 by a real number say k and add the corresponding product to row 1.

  $\begin{array}{c}{D}_1=\left|\begin{array}{cc}a+bk& c+kd\\ b& d\end{array}\right|\\ =\left(a+bk\right)d-\left(c+kd\right)b\\ =ad-bdk-cb-bdk\\ =ad-bc\end{array}$

Observe that $D=D_1$ .

Hence, it is proved that the value of new second order determinant is same as the original one if all entries of either a row or a column are multiplied by a real number k , and the resultant products are added to corresponding entries of either another row or another column