The determinant of matrices is often used in calculus, linear algebra, and geometry at a higher level. Outside of academia, computer graphics engineers and programmers use matrices and their determinants all the time. If you already know how to determine the determinant of a 2x2-order matrix, you just need to learn when to use addition, subtraction, and times to determine the determinant of a 3x3-order matrix.
Step
Part 1 of 2: Determining the Determinants
Write your 3 x 3 order matrix. We will start with a matrix A of order 3x3 and try to find the determinant |A|. Below is the general form of matrix notation we will use and an example of our matrix:
| a11 | a12 | a13 | 1 | 5 | 3 | |||
| M | = | a21 | a22 | a23 | = | 2 | 4 | 7 |
| a31 | a32 | a33 | 4 | 6 | 2 |
Step 1. Select a row or column
Make your selection the reference row or column. Whichever you choose, you'll still get the same answer. Temporarily select the first row. We'll give you some suggestions for choosing the easiest-to-calculate option in the next section.
Select the first row of the sample matrix A. Circle the number 1 5 3. In common notation, circle a11 a12 a13.
Step 2. Cross out the row and column of your first element
Look at the row or column you circled and select the first element. Cross out the rows and columns. There will only be 4 numbers left untouched. Make these 4 numbers a 2 x 2 order matrix.
- In our example, our reference row is 1 5 3. The first element is in the 1st row and 1st column. Cross out the entire 1st row and 1st column. Write the remaining elements into a 2 x 2 matrix:
- 1 5 3
- 2 4 7
- 4 6 2
Step 3. Determine the determinant of the 2 x 2 order matrix
Remember, determine the determinant of the matrix [ac bd] by ad - bc. You may also have learned to determine the determinant of a matrix by drawing an X between a 2 x 2 matrix. Multiply the two numbers connected by the line / of X. Then, subtract the number of times the two numbers connected by the line / are. Use this formula to calculate the determinant of a 2 x 2 matrix.
- In the example, the determinant of the matrix [46 72] = 4*2 - 7*6 = -34.
- This determinant is called minor of the elements you selected in the initial matrix. In this case, we have just found the minor of a11.
Step 4. Multiply the found number by the element you selected
Remember, you have selected elements from the reference row (or column) when you decided which rows and columns to strike out. Multiply this element by the determinant of the 2 x 2 matrix you've found.
In the example, we choose a11 which is 1. Multiply this number by -34 (the determinant of the 2 x 2 matrix) to get 1*-34 = -34.
Step 5. Determine the symbol of your answer
The next step is that you have to multiply your answer by 1 or -1 to get cofactor of the element you selected. The symbol you use depends on where the elements are in the 3 x 3 matrix. Remember, this symbol table is used to determine your element's multiplier:
- + - +
- - + -
- + - +
- Because we choose a11 which is marked a +, we will multiply the number by +1 (or in other words, don't change it). The answer that appears will be the same, namely -34.
- Another way to define a symbol is to use the formula (-1) i+j where i and j are row and column elements.
Step 6. Repeat this process for the second element in your reference row or column
Return to the original 3 x 3 matrix that you circled the row or column in earlier. Repeat the same process with the element:
-
Cross out the row and column of the element.
In this case, select the element a12 (which is worth 5). Cross out the 1st row (1 5 3) and the 2nd column (5 4 6).
-
Turn the remaining elements into a 2x2 matrix.
In our example, the 2x2 order matrix for the second element is [24 72].
-
Determine the determinant of this 2x2 matrix.
Use the ad - bc formula. (2*2 - 7*4 = -24)
-
Multiply by the elements in your chosen 3x3 matrix.
-24 * 5 = -120
-
Decide whether to multiply the above result by -1 or not.
Use a table of symbols or formulas (-1)ij. Select element a12 symbolized – in the symbol table. Replace our answer symbol with: (-1)*(-120) = 120.
Step 7. Repeat the same process for the third element
You have one more cofactor to determine the determinant. Count i for the third element in your reference row or column. Here is a quick way to calculate the cofactor a13 in our example:
- Cross out the 1st row and 3rd column to get [24 46].
- The determinant is 2*6 - 4*4 = -4.
- Multiply by element a13: -4 * 3 = -12.
- Element a13 symbol + in the symbol table, so the answer is -12.
Step 8. Add up the results of your three counts
This is the last step. You have calculated three cofactors, one for each element in a row or column. Add up those results and you'll find the determinant of a 3 x 3 matrix.
In the example, the determinant of the matrix is -34 + 120 + -12 = 74.
Part 2 of 2: Making Problem Solving Easier
Step 1. Select the row or column of references that have the most 0s
Remember, you can select any row or column you want. Whichever you choose, the answer will be the same. If you select a row or column with the number 0, you only need to calculate the cofactor with elements that are not 0 because:
- For example, select the 2nd row which has the element a21, a22, fund23. To solve this problem, we will use 3 different 2 x 2 matrices, let's say A21, A22, You23.
- The determinant of the 3x3 matrix is a21|A21| - a22|A22| + a23|A23|.
- If a22 fund23 value 0, the existing formula will be a21|A21| - 0*|A22| + 0*|A23| = a21|A21| - 0 + 0 = a21|A21|. Therefore, we will only calculate the cofactor of one element only.
Step 2. Use extra rows to make matrix problems easier
If you take values from one row and add them to another row, the determinant of the matrix will not change. The same is true for columns. You can do this repeatedly or multiply by a constant before adding it to get as many 0's in the matrix as possible. This can save a lot of time.
- For example, you have a matrix with 3 rows: [9 -1 2] [3 1 0] [7 5 -2]
- To eliminate the number 9 which is in position a11, you can multiply the value in the 2nd row by -3 and add the result to the first row. Now, the new first line is [9 -1 2] + [-9 -3 0] = [0 -4 2].
- The new matrix has rows [0 -4 2] [3 1 0] [7 5 -2]. Use the same trick on columns to make a12 be the number 0.
Step 3. Use the quick method for triangular matrices
In this special case, the determinant is the product of the elements on the main diagonal, of a11 at the top left to a33 at the bottom right of the matrix. This matrix is still a 3x3 matrix, but the "triangle" matrix has a special pattern of numbers that are not 0:
- Upper triangular matrix: All elements that are not 0 are on or above the main diagonal. All numbers below the main diagonal are 0.
- Bottom triangular matrix: All elements that are not 0 are on or below the main diagonal.
- Diagonal matrix: All elements that are not 0 are on the main diagonal (the subset of the above types of matrices).
Tips
- If all the elements in a row or column are 0, the determinant of the matrix is 0.
- This method can be used for all sizes of quadratic matrices. For example, if you use this method for a matrix of order 4x4, your "strike" will leave a matrix of order 3x3 whose determinant can be determined by following the steps above. Remember, doing this can be boring!
