Question

The determinant of the matrix$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ is___

Answer 1

We will evaluate this using cofactors. Let’s evaluate it along the first row.

Expansion by first row

$\left|\begin{array}{ccc}1& \ast & \ast \\ \ast & 5& 6\\ \ast & 8& 9\end{array}\right|\left|\begin{array}{ccc}\ast & 2& \ast \\ 4& \ast & 6\\ 7& \ast & 9\end{array}\right|\left|\begin{array}{ccc}\ast & \ast & 3\\ 4& 5& \ast \\ 7& 8& \ast \end{array}\right|$

Here what we will do is that we will multiply each element of the first row with its cofactor which is
$(-1{)}^{i+j}\times$ (determinant of the matrix left after deleting the row and column containing that element, as shown in the 3 matrices above) Here i and j are row and column of the element.

So det $A=1\left|\begin{array}{cc}5& 6\\ 8& 9\end{array}\right|-2\left|\begin{array}{cc}4& 6\\ 7& 9\end{array}\right|+3\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|$
$=(5\cdot 9–6\cdot 8)-2(4\cdot 9-7\cdot 6)+3(4\cdot 8-7\cdot 5)=0$ which is the correct answer