Refresher - Determinants

Determinants

Each square matrix has a determinant which is a number associated with the matrix. The determinant of $[\begin{matrix} a & b \\ c & d \end{matrix}]$ can be denoted as $| \begin{matrix} a & b \\ c & d \end{matrix} |$ or det $[\begin{matrix} a & b \\ c & d \end{matrix}]$ . The value of or det $[\begin{matrix} a & b \\ c & d \end{matrix}]$ is ad-bc.

For determinants larger than a second-order (2 x 2) we use a method that utilizes minors. A minor of an element can be found by deleting the row and column containing the element.

Example:

$[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}]$ The minor of $a_{1}$ is $[\begin{matrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{matrix}]$

One method of evaluating a determinant is by expanding the determinant by its minors. In the example below of a third-order determinant, elements of the first row were crossed out. Note that the signs between the terms are alternating. An easy way to determine if the element is + or – is to add the row and column number of the element. If the sum is even, the sign is positive; if the sum is odd, the sign is negative. In the example below, $a_{1}$ $a_{1}$ is in row 1 + column 1 = 2. The sum is even, so the sign is positive. $a_{1}$

$| \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} | = a_{1} | \begin{matrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{matrix} | - b_{1} | \begin{matrix} a_{2} & c_{2} \\ a_{3} & c_{3} \end{matrix} | + c_{1} | \begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix} |$

Example:

$| \begin{matrix} 3 & 2 & 1 \\ 2 & 0 & - 1 \\ 3 & 2 & 4 \end{matrix} | = 3 | \begin{matrix} 0 & - 1 \\ 2 & 4 \end{matrix} | - 2 | \begin{matrix} 2 & - 1 \\ 3 & 4 \end{matrix} | + 1 | \begin{matrix} 2 & 0 \\ 3 & 2 \end{matrix} |$

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