1.1: Definition of a Matrix
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An \(m\)-by-\(n\) matrix is a rectangular array of numbers (or other mathematical objects) with \(m\) rows and \(n\) columns. For example, a two-by-two matrix \(\text{A}\), with two rows and two columns, looks like
\[\text{A}=\left(\begin{array}{cc}a&b\\c&d\end{array}\right).\nonumber \]
(Sometimes brackets are used instead of parentheses.) The first row has elements \(a\) and \(b\), the second row has elements \(c\) and \(d\). The first column has elements \(a\) and \(c\); the second column has elements \(b\) and \(d\). As further examples, \(2\)-by-\(3\) and \(3\)-by-\(2\) matrices look like
\[\text{B}=\left(\begin{array}{ccc}a&b&c\\d&e&f\end{array}\right),\quad \text{C}=\left(\begin{array}{cc}a&b\\c&d\\e&f\end{array}\right).\nonumber \]
Of special importance are the so-called row matrices and column matrices. These matrices are also called row vectors and column vectors. The row vector is in general \(1\)-by-\(n\) and the column vector is \(n\)-by-\(1\). For example, when \(n = 3\), we would write
\[\text{v}=\left(\begin{array}{ccc}a&b&c\end{array}\right)\nonumber \]
as a row vector, and
\[\text{v}=\left(\begin{array}{c}a\\b\\c\end{array}\right)\nonumber \]
as a column vector.