1.3: The Identity Matrix and the Zero Matrix
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Two special matrices are the identity matrix, denoted by \(\text{I}\), and the zero matrix, denoted simply by \(0\). The zero matrix can be \(m\)-by-\(n\) and is a matrix consisting of all zero elements. The identity matrix is a square matrix. If \(\text{A}\) and \(\text{I}\) are of the same size, then the identity matrix satisfies
\[\text{AI}=\text{IA}=\text{A},\nonumber \]
and plays the role of the number one in matrix multiplication. The identity matrix consists of ones along the diagonal and zeros elsewhere. For example, the \(3\)-by-\(3\) zero and identity matrices are given by
\[0=\left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right),\quad\text{I}=\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right),\nonumber \]
and it is easy to check that
\[\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right)=\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right)\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)=\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right).\nonumber \]
Although strictly speaking, the symbols \(0\) and \(\text{I}\) represent different matrices depending on their size, we will just use these symbols and leave their exact size to be inferred.