Calculate the determinant of I = [[2, 3, 1], [1, 0, 2], [0, 1, 3]]. (2015)

Calculate the determinant of I = [[2, 3, 1], [1, 0, 2], [0, 1, 3]]. (2015)

Step 1: Write down the matrix I: [[2, 3, 1], [1, 0, 2], [0, 1, 3]].
Step 2: Identify the elements of the matrix: a = 2, b = 3, c = 1, d = 1, e = 0, f = 2, g = 0, h = 1, i = 3.
Step 3: Use the formula for the determinant of a 3x3 matrix: Det(I) = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 4: Substitute the values into the formula: Det(I) = 2(0*3 - 2*1) - 3(1*3 - 2*0) + 1(1*1 - 0*0).
Step 5: Calculate each part: 0*3 = 0, 2*1 = 2, so 0 - 2 = -2; 1*3 = 3, 2*0 = 0, so 3 - 0 = 3; 1*1 = 1, 0*0 = 0, so 1 - 0 = 1.
Step 6: Substitute these results back into the determinant formula: Det(I) = 2(-2) - 3(3) + 1(1).
Step 7: Calculate: 2 * -2 = -4, -3 * 3 = -9, and 1 * 1 = 1.
Step 8: Combine these results: -4 - 9 + 1 = -12.
Step 9: The final result is Det(I) = -12.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the formula involving minors and cofactors.
Matrix Properties – Understanding properties of determinants, such as linearity and how they relate to the invertibility of matrices.