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Determinants

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• 1. Determinants
When taking the determinant of a matrix, we can “expand” by any row or any column
• 2. Step 1: Choose a row or column
• 3. Step 1: Choose a row or column
Let’s choose the first row
• 4. *Always remember that signs alternate. This will make more sense later.
• 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
So we get 2 times the determinant of the matrix of non-circled numbers
• 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
So we get 2 times the determinant of the matrix of the numbers in the rectangle
• 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
2*[(-5*6)-(8*0)] = 2*(-30-0) = -60
• 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
2*[(-5*6)-(8*0)] = 2*(-30-0) = -60
THIS IS NOT THE FINAL ANSWER!
• 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
This is where the alternating signs come in
• 10. This is where the alternating signs come in
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
• 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
-(-3)*[(5*-5)-(8*1)] = 3*(-25-8) = -99
• 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
-(-3)*[(5*-5)-(8*1)] = 3*(-25-8) = -99
THIS IS NOT THE FINAL ANSWER!
• 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
11*[(5*0)-(6*1) = 11*(0-6) = -66
• 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
11*[(5*0)-(6*1) = 11*(0-6) = -66
THIS IS NOT THE FINAL ANSWER!
• 15. We now have the three parts of our answer. To find the actual answer, we take the sum of our three parts
-60 -99 -66 = -225
So the determinant of our matrix is -225