Question

Show without computations that the following determinant is equal to zero. Hint: Consider the effect of interchanging rows and columns.

$\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& 4\\ 3& -4& 0\end{array}\right|$

Short answer

It has been proved that the determinant is zero.

Step-by-step solution

Results used

Facts used about determinants:

1. If each element of one row of a determinant is multiplied by a number k, the determinant becomes k times.
2. The value of a determinant remains the same if rows are written as columns and columns as rows.

Show that the determinant is zero

Consider,$D=\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& 4\\ 3& -4& 0\end{array}\right|$

Write each row as a column and vice versa, using fact 2 as:

$D=\left|\begin{array}{ccc}0& -2& 3\\ 2& 0& -4\\ -3& 4& 0\end{array}\right|$

Multiply each row by -1. So Using fact 1, the determinant is

$$\begin{gathered} D={\left(-1\right)}^3\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& -4\\ 3& -4& 0\end{array}\right| \\ =-D \end{gathered}$$

Thus,

$$\begin{gathered} D+D=0 \\ 2D=0 \\ D=0 \end{gathered}$$

Hence, determinant is zero.