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Chapter 6: Problem 57

Explain how to evaluate a third-order determinant.

Short Answer

Expert verified
The determinant of a 3x3 matrix A is calculated using the formula \[ |A|= (a_{11}a_{22}a_{33})+(a_{12}a_{23}a_{31})+(a_{13}a_{21}a_{32})-(a_{13}a_{22}a_{31})-(a_{11}a_{23}a_{32})-(a_{12}a_{21}a_{33}) \], which is derived from finding the difference between the sum of the products of the diagonals from left to right and right to left.

Step by step solution

01

Define the 3x3 Matrix

Firstly, a 3x3 matrix has to be defined. For instance, the matrix could be: \[ A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} \]
02

Calculate the Diagonals from Left to Right

In this step, multiply elements in a diagonal from left to right. It's done by adding a copy of the first two columns to the matrix and performing multiplication along the line. So it becomes: \[ diag_{left}= (a_{11}a_{22}a_{33})+(a_{12}a_{23}a_{31})+(a_{13}a_{21}a_{32}) \]
03

Calculate the Diagonals from Right to Left

Next, multiply elements in a diagonal from right to left. Similar to step 2, add a copy of the first two columns to the matrix and perform multiplication along the line. It will look like: \[ diag_{right}= (a_{13}a_{22}a_{31})+(a_{11}a_{23}a_{32})+(a_{12}a_{21}a_{33}) \]
04

Compute the Difference

Subtract the result of Step 3 from the result of Step 2. This difference is the determinant of the 3x3 matrix, which is represented as: \[ |A|=diag_{left}-diag_{right} \]

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