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Determinants

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Determinants

1. 1. Determinants<br />When taking the determinant of a matrix, we can “expand” by any row or any column<br />
2. 2. Step 1: Choose a row or column<br />
3. 3. Step 1: Choose a row or column<br />Let’s choose the first row<br />
4. 4. *Always remember that signs alternate. This will make more sense later.<br />
5. 5. Step 2: Take the first number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />So we get 2 times the determinant of the matrix of non-circled numbers<br />
6. 6. Step 2: Take the first number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />So we get 2 times the determinant of the matrix of the numbers in the rectangle<br />
7. 7. Step 2: Take the first number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />2*[(-5*6)-(8*0)] = 2*(-30-0) = -60<br />
8. 8. Step 2: Take the first number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />2*[(-5*6)-(8*0)] = 2*(-30-0) = -60<br />THIS IS NOT THE FINAL ANSWER!<br />
9. 9. Step 3: Take the next number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />This is where the alternating signs come in<br />
10. 10. This is where the alternating signs come in<br />Since the circled number is in the same position as a negative sign, we multiply it by negative 1 before multiplying it by the determinant of the 2x2 matrix in Step 4<br />
11. 11. Step 3: Take the next number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />-(-3)*[(5*-5)-(8*1)] = 3*(-25-8) = -99<br />
12. 12. Step 3: Take the next number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />-(-3)*[(5*-5)-(8*1)] = 3*(-25-8) = -99<br />THIS IS NOT THE FINAL ANSWER!<br />
13. 13. Step 4: Take the last number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />11*[(5*0)-(6*1) = 11*(0-6) = -66<br />
14. 14. Step 4: Take the last number in our row and multiply it by the determinant of the matrix left after eliminating all adjacent numbers<br />11*[(5*0)-(6*1) = 11*(0-6) = -66<br />THIS IS NOT THE FINAL ANSWER!<br />
15. 15. We now have the three parts of our answer. To find the actual answer, we take the sum of our three parts<br />-60 -99 -66 = -225<br />So the determinant of our matrix is -225<br />