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Chapter 6: Problem 53

Explain how to evaluate a third-order determinant.

Short Answer

Expert verified
The determinant of a 3x3 matrix \[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\] can be calculated using the formula \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\]. Replace each variable in the formula with the corresponding element from the matrix, compute the result to find the determinant.

Step by step solution

01

Identify the 3x3 matrix

Write down your 3x3 matrix. It should contain nine elements arranged in three rows and three columns. It can be denoted as \[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\]
02

Expansion by Minors

Expand the determinant across a row or a column. The expansion is generally done across the top row but it can be done using any row or column. This gives the formula for the determinant as \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\]
03

Calculate the Result

Simply substitute the values from your matrix into the formula. Evaluate the expressions in parentheses first, according to the order of operations, then multiply and add/subtract as indicated to find the determinant of the 3x3 matrix.

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Most popular questions from this chapter

Consider the system $$ \begin{array}{l}a_{1} x+b_{1} y=c_{1} \\\a_{2} x+b_{2} y=c_{2}\end{array} $$ Use Cramer's rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original system.

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{rrrr} 2 & -3 & 1 & -1 \\ 1 & 1 & -2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 2 \\ -1 & 1 \\ 5 & 4 \\ 10 & 5 \end{array}\right] $$

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&4 y=16-3 x\\\&6 x=32-8 y\end{aligned} $$

If two matrices can be multiplied, describe how to determine the order of the product.

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices\(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work. Use determinants to find the area of the triangle whose vertices are \((3,-5),(2,6),\) and \((-3,5)\).
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