Find the determinant of the matrix D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

Find the determinant of the matrix D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

Step 1: Write down the matrix D: [[3, 2, 1], [1, 0, 2], [2, 1, 3]].
Step 2: Use the formula for the determinant of a 3x3 matrix: det(D) = a(ei - fh) - b(di - fg) + c(dh - eg), where D = [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Identify the elements from the matrix: a = 3, b = 2, c = 1, d = 1, e = 0, f = 2, g = 2, h = 1, i = 3.
Step 4: Calculate ei - fh: (0 * 3) - (2 * 1) = 0 - 2 = -2.
Step 5: Calculate di - fg: (1 * 3) - (2 * 2) = 3 - 4 = -1.
Step 6: Calculate dh - eg: (1 * 1) - (0 * 2) = 1 - 0 = 1.
Step 7: Substitute these values into the determinant formula: det(D) = 3 * (-2) - 2 * (-1) + 1 * 1.
Step 8: Calculate: det(D) = -6 + 2 + 1 = -3.
Step 9: Since the determinant is not zero, the final result is det(D) = -3.

Determinant Calculation – Understanding how to compute the determinant of a 3x3 matrix using methods like the rule of Sarrus or cofactor expansion.
Matrix Properties – Knowledge of properties of determinants, including how they relate to the invertibility of matrices.