Find the value of the determinant: | x 1 2 | | 3 x 4 | | 5 6 x | when x = 1.

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Q: Find the value of the determinant: | x 1 2 | | 3 x 4 | | 5 6 x | when x = 1.

Solution: Substituting x = 1 gives the determinant | 1 1 2 | | 3 1 4 | | 5 6 1 | = 6.

Steps: 11

Step 1: Write down the determinant with the variable x: | x 1 2 | | 3 x 4 | | 5 6 x |.
Step 2: Substitute x = 1 into the determinant: | 1 1 2 | | 3 1 4 | | 5 6 1 |.
Step 3: Now, calculate the determinant of the 3x3 matrix | 1 1 2 | | 3 1 4 | | 5 6 1 |.
Step 4: Use the formula for the determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 5: Identify the elements from the matrix: a=1, b=1, c=2, d=3, e=1, f=4, g=5, h=6, i=1.
Step 6: Calculate ei - fh: (1*1) - (4*6) = 1 - 24 = -23.
Step 7: Calculate di - fg: (3*1) - (4*5) = 3 - 20 = -17.
Step 8: Calculate dh - eg: (3*6) - (1*5) = 18 - 5 = 13.
Step 9: Substitute these values into the determinant formula: det(A) = 1*(-23) - 1*(-17) + 2*(13).
Step 10: Calculate: det(A) = -23 + 17 + 26 = 20.
Step 11: The final value of the determinant when x = 1 is 20.