If \( D = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{vmatrix} \), find \( D \).

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Q: If \( D = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{vmatrix} \), find \( D \).

Solution: Calculating gives \( 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = -4 + 24 + 1 = 21 \).

Steps: 9

Step 1: Write down the determinant formula for a 3x3 matrix. The determinant D of a matrix is calculated as follows: D = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is: [[a, b, c], [d, e, f], [g, h, i]].
Step 2: Identify the elements of the matrix. For the given matrix D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]], we have: a = 2, b = 3, c = 1, d = 1, e = 0, f = 2, g = 4, h = 1, i = 0.
Step 3: Calculate the first part of the determinant: ei - fh. Here, e = 0, i = 0, f = 2, h = 1. So, ei - fh = (0*0) - (2*1) = 0 - 2 = -2.
Step 4: Calculate the second part of the determinant: di - fg. Here, d = 1, i = 0, f = 2, g = 4. So, di - fg = (1*0) - (2*4) = 0 - 8 = -8.
Step 5: Calculate the third part of the determinant: dh - eg. Here, d = 1, h = 1, e = 0, g = 4. So, dh - eg = (1*1) - (0*4) = 1 - 0 = 1.
Step 6: Substitute these values back into the determinant formula: D = 2(-2) - 3(-8) + 1(1).
Step 7: Calculate each term: 2(-2) = -4, -3(-8) = 24, and 1(1) = 1.
Step 8: Add these results together: -4 + 24 + 1 = 21.
Step 9: Therefore, the value of D is 21.