Evaluate the determinant Δ=1−14231400.
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Step-by-Step Solution
Step 1: Choose a row or column for expansion
To evaluate a determinant, we can use the cofactor expansion method. It's often easiest to expand along a row or column that contains the most zeros, as this simplifies calculations. In this matrix, the third column has two zeros, so we will expand along the third column.
Step 2: Apply cofactor expansion along the third column
The determinant is calculated as the sum of the products of each element in the chosen column (or row) with its corresponding cofactor. The cofactor Cij is given by (−1)i+jMij, where Mij is the minor determinant obtained by removing the i -th row and j -th column. Since the elements in the second and third rows of the third column are zero, their contributions to the determinant will be zero.
Step 3: Calculate the cofactor C13
For the element a13=4, its cofactor C13 is (−1)1+3 times the determinant of the 2×2 matrix obtained by removing the first row and third column. This 2×2 matrix is (−1431).
Step 4: Evaluate the 2×2 minor determinant
The determinant of a 2×2 matrix (acbd) is ad - bc. Applying this formula to our minor determinant, we get (−1)(1)−(3)(4), which simplifies to −1−12=−13.
Step 5: Substitute the minor determinant back into the cofactor
Now we substitute the value of the 2×2 determinant back into the cofactor formula. Since (−1)1+3=(−1)4=1, the cofactor C13 is 1×(−13)=−13.
Step 6: Calculate the final determinant
Finally, we substitute the value of C13 back into the expansion formula: Δ=4⋅C13. This gives us 4⋅(−13)=−52.