Problem 57 - College Algebra

Question

Explain how to evaluate a third-order determinant.

Step-by-Step Solution

Answer

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 1: Define the 3x3 Matrix

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

Step 2: 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: \[ diag_{left}= (a_{11}a_{22}a_{33})+(a_{12}a_{23}a_{31})+(a_{13}a_{21}a_{32}) \]

Step 3: 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: \[ diag_{right}= (a_{13}a_{22}a_{31})+(a_{11}a_{23}a_{32})+(a_{12}a_{21}a_{33}) \]

Step 4: 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: \[ |A|=diag_{left}-diag_{right} \]