Difference between revisions of "Find the Determinant of a 3X3 Matrix"
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| − | The determinant of a matrix is frequently used in calculus, linear algebra, and higher level geometry. Outside the academic world, engineers and computer graphics programmers use matrices and their determinants all the time.<ref>http://www.decodedscience.com/practical-uses-matrix-mathematics/40494</ref> To find the determinant of a 3x3 matrix, read this | + | The determinant of a matrix is frequently used in calculus, linear algebra, and higher level geometry. Outside the academic world, engineers and computer graphics programmers use matrices and their determinants all the time.<ref name="rf16605">http://www.decodedscience.com/practical-uses-matrix-mathematics/40494</ref> To find the determinant of a 3x3 matrix, read this article. |
== 10 Second Summary == | == 10 Second Summary == | ||
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9. Add your three results together. [[#step_1_9|↓]]<br> | 9. Add your three results together. [[#step_1_9|↓]]<br> | ||
| − | [[Category:Linear Algebra]] | + | [[Category: Linear Algebra]] |
== Steps == | == Steps == | ||
===Finding the Determinant=== | ===Finding the Determinant=== | ||
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#*In our example, our reference row is 1 5 3. The <font color="red">first element</font> is in row 1 and column 1. Cross out all of row 1 and column 1. Write the remaining elements as a <font color="blue">2 x 2 matrix</font>: | #*In our example, our reference row is 1 5 3. The <font color="red">first element</font> is in row 1 and column 1. Cross out all of row 1 and column 1. Write the remaining elements as a <font color="blue">2 x 2 matrix</font>: | ||
#*<s><font color="red"> 1 </font> 5 3</s><br><s> 2 </s> <font color="blue">'''4 7'''</font><br><s> 4 </s> <font color="blue">'''6 2'''</font> | #*<s><font color="red"> 1 </font> 5 3</s><br><s> 2 </s> <font color="blue">'''4 7'''</font><br><s> 4 </s> <font color="blue">'''6 2'''</font> | ||
| − | #Find the determinant of the 2 x 2 matrix. Remember, the matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> has a determinant of '''ad - bc'''.<ref>https://www.khanacademy.org/math/precalculus/precalc-matrices/inverting_matrices/v/finding-the-determinant-of-a-2x2-matrix</ref> You may have learned this by drawing an X across the 2 x 2 matrix. Multiply the two numbers connected by the \ of the X. Then subtract the product of the two numbers connected by the /. Use this formula to calculate the determinate of the matrix you just found. | + | #Find the determinant of the 2 x 2 matrix. Remember, the matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> has a determinant of '''ad - bc'''.<ref name="rf16606">https://www.khanacademy.org/math/precalculus/precalc-matrices/inverting_matrices/v/finding-the-determinant-of-a-2x2-matrix</ref> You may have learned this by drawing an X across the 2 x 2 matrix. Multiply the two numbers connected by the \ of the X. Then subtract the product of the two numbers connected by the /. Use this formula to calculate the determinate of the matrix you just found. |
#*In our example, the determinant of the matrix <font color="blue"><math>\begin{pmatrix}4 & 7 \\ 6 & 2 \end{pmatrix}</math></font> = 4 * 2 - 7 * 6 = '''-34'''. | #*In our example, the determinant of the matrix <font color="blue"><math>\begin{pmatrix}4 & 7 \\ 6 & 2 \end{pmatrix}</math></font> = 4 * 2 - 7 * 6 = '''-34'''. | ||
| − | #*This determinant is called the '''minor''' of the element we chose in our original matrix.<ref>https://people.richland.edu/james/lecture/m116/matrices/determinant.html</ref> In this case, we just found the minor of <font color="red">a<sub>11</sub></font>. | + | #*This determinant is called the '''minor''' of the element we chose in our original matrix.<ref name="rf16607">https://people.richland.edu/james/lecture/m116/matrices/determinant.html</ref> In this case, we just found the minor of <font color="red">a<sub>11</sub></font>. |
#Multiply the answer by your chosen element. Remember, you selected an element from your reference row (or column) when you decided which row and column to cross out. Multiply this element by the determinant you just calculated for the 2x2 matrix. | #Multiply the answer by your chosen element. Remember, you selected an element from your reference row (or column) when you decided which row and column to cross out. Multiply this element by the determinant you just calculated for the 2x2 matrix. | ||
#*In our example, we selected a<sub>11</sub>, which had a value of 1. Multiply this by -34 (the determinant of the 2x2) to get 1*-34 = '''-34'''. | #*In our example, we selected a<sub>11</sub>, which had a value of 1. Multiply this by -34 (the determinant of the 2x2) to get 1*-34 = '''-34'''. | ||
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#*<font color="red">+</font> - +<br>- + -<br>+ - + | #*<font color="red">+</font> - +<br>- + -<br>+ - + | ||
#*Since we chose <font color="red">a<sub>11</sub></font>, marked with a +, we multiply the number by +1. (In other words, leave it alone.) The answer is still <font color="orange">'''-34'''</font>. | #*Since we chose <font color="red">a<sub>11</sub></font>, marked with a +, we multiply the number by +1. (In other words, leave it alone.) The answer is still <font color="orange">'''-34'''</font>. | ||
| − | #*Alternatively, you can find the sign with the formula (-1)<sup>''i+j''</sup>, where ''i'' and ''j'' are the element's row and column.<ref>http://www.math.rutgers.edu/~cherlin/Courses/250/Lectures/250L12.html</ref> | + | #*Alternatively, you can find the sign with the formula (-1)<sup>''i+j''</sup>, where ''i'' and ''j'' are the element's row and column.<ref name="rf16608">http://www.math.rutgers.edu/~cherlin/Courses/250/Lectures/250L12.html</ref> |
#Repeat this process for the second element in your reference row or column. Return to the original 3x3 matrix, with the row or column you circled earlier. Repeat the same process with this element: | #Repeat this process for the second element in your reference row or column. Return to the original 3x3 matrix, with the row or column you circled earlier. Repeat the same process with this element: | ||
#*'''Cross out the row and column of that element.''' In our case, select element a<sub>12</sub> (with a value of 5). Cross out row one (1 5 3) and column two <font size=2><math>\begin{pmatrix}5 \\ 4 \\ 6\end{pmatrix}</math></font>. | #*'''Cross out the row and column of that element.''' In our case, select element a<sub>12</sub> (with a value of 5). Cross out row one (1 5 3) and column two <font size=2><math>\begin{pmatrix}5 \\ 4 \\ 6\end{pmatrix}</math></font>. | ||
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#*In order to cancel out the 9 in position a<sub>11</sub>, we can multiply the second row by -3 and add the result to the first. The new first row is <nowiki>[9 -1 2] + [-9 -3 0] = [0 -4 2]</nowiki>. | #*In order to cancel out the 9 in position a<sub>11</sub>, we can multiply the second row by -3 and add the result to the first. The new first row is <nowiki>[9 -1 2] + [-9 -3 0] = [0 -4 2]</nowiki>. | ||
#*The new matrix is <math> \begin{pmatrix} 0 & -4 & 2 \\ 3 & 1 & 0 \\ 7 & 5 & -2 \end{pmatrix}</math> Try to use the same trick with columns to turn a<sub>12</sub> into a 0 as well. | #*The new matrix is <math> \begin{pmatrix} 0 & -4 & 2 \\ 3 & 1 & 0 \\ 7 & 5 & -2 \end{pmatrix}</math> Try to use the same trick with columns to turn a<sub>12</sub> into a 0 as well. | ||
| − | #Learn the shortcut for triangular matrices. In these special cases, the determinant is simply the product of the elements along the main diagonal, from a<sub>11</sub> in the top left to a<sub>33</sub> in the lower right. We're still talking about 3x3 matrices, but "triangular" ones have special patterns of ''nonzero'' values:<ref | + | #Learn the shortcut for triangular matrices. In these special cases, the determinant is simply the product of the elements along the main diagonal, from a<sub>11</sub> in the top left to a<sub>33</sub> in the lower right. We're still talking about 3x3 matrices, but "triangular" ones have special patterns of ''nonzero'' values:<ref name="rf16607" /> |
#*Upper triangular matrix: All the non-zero elements are on or above the main diagonal. Everything below is a zero. | #*Upper triangular matrix: All the non-zero elements are on or above the main diagonal. Everything below is a zero. | ||
#*Lower triangular matrix: All the non-zero elements are on or below the main diagonal. | #*Lower triangular matrix: All the non-zero elements are on or below the main diagonal. | ||
#*Diagonal matrix: All the non-zero elements are on the main diagonal. (A subset of the above.) | #*Diagonal matrix: All the non-zero elements are on the main diagonal. (A subset of the above.) | ||
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== Tips == | == Tips == | ||
