16.6 Higher-order Determinants

Minors • Expansion by Minors • Properties of Determinants • Solving Systems of Equations

In Chapter 5, we limited our discussion of determinants to those of the second and third orders. We now show some methods of evaluating higher-order determinants, and use these methods to solve systems of equations.

From Section 5.7, we recall the definition of a third-order determinant. By rearranging the terms and factoring out $a_{1}, - a_{2},$ and $a_{3},$ we have

\begin{array}{rcl} | \begin{array}{l} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | & = & a_{1} b_{2} c_{3} + a_{3} b_{1} c_{2} + a_{2} b_{3} c_{1} - a_{3} b_{2} c_{1} - a_{1} b_{3} c_{2} - a_{2} b_{1} c_{3} \\ = & a_{1} (b_{2} c_{3} - b_{3} c_{2}) - a_{2} (b_{1} c_{3} - b_{3} c_{1}) + a_{3} (b_{1} c_{2} - b_{2} c_{1}) \\ = & a_{1} | \begin{array}{l} b_{2} & c_{2} \\ b_{3} & c_{3} \end{array} | - a_{2} | \begin{array}{l} b_{1} & c_{1} \\ b_{3} & c_{3} \end{array} | + a_{3} | \begin{array}{l} b_{1} & c_{1} \\ b_{2} & c_{2} \end{array} | \end{array}

(16.12)

In Eq. (16.12), the third-order determinant is expanded as products ...

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Basic Technical Mathematics, 11th Edition by Allyn J. Washington, Richard Evans

16.6 Higher-order Determinants

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