4.1: Two-by-Two and Three-by-Three Determinants
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Our first introduction to determinants was the definition for the general two-by-two matrix
\[\text{A}=\left(\begin{array}{cc}a&b\\c&d\end{array}\right):\quad\det\text{A}=ad-bc.\nonumber \]
Other widely used notations for the determinant include
\[\det\text{A}=\det\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=|\text{A}|=\left|\begin{array}{cc}a&b\\c&d\end{array}\right|.\nonumber \]
By explicit construction, we have seen that a two-by-two matrix \(\text{A}\) is invertible if and only if \(\det\text{A}\neq= 0\). If a square matrix \(\text{A}\) is invertible, then the equation \(\text{Ax} = \text{b}\) has the unique solution \(\text{x} = \text{A}^{−1}\text{b}\). But if \(\text{A}\) is not invertible, then \(\text{Ax} = \text{b}\) may have no solution or an infinite number of solutions. When \(\det \text{A} = 0\), we say that \(\text{A}\) is a singular matrix.
Here, we would like to extend the definition of the determinant to an \(n\)-by-\(n\) matrix. Before we do so, let us display the determinant for a three-by-three matrix. We consider the system of equations \(\text{Ax} = 0\) and find the condition for which \(\text{x} = 0\) is the only solution. This condition must be equivalent to \(\det \text{A}\neq 0\). With
\[\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)\left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right)=0,\nonumber \]
one can do the messy algebra of elimination to solve for \(\text{x}_1\), \(\text{x}_2\), and \(\text{x}_3\). One finds that \(\text{x}_1 = \text{x}_2 = \text{x}_3 = 0\) is the only solution when \(\det \text{A}\neq 0\), where the definition is given by
\[\det\text{A}=aei+bfg+cdh-ceg-bdi-afh.\label{eq:1} \]
A way to remember this result for the three-by-three matrix is by the following picture:
The matrix \(\text{A}\) is periodically extended two columns to the right, drawn explicitly here but usually only imagined. Then the six terms comprising the determinant are made evident, with the lines slanting down towards the right getting the plus signs and the lines slanting down towards the left getting the minus signs. Unfortunately, this mnemonic is only valid for three-by-three matrices.