If I = [[1, 0, 2], [0, 1, 3], [1, 0, 4]], find det(I). (2021)

If I = [[1, 0, 2], [0, 1, 3], [1, 0, 4]], find det(I). (2021)

Step 1: Identify the matrix I. The matrix I is [[1, 0, 2], [0, 1, 3], [1, 0, 4]].
Step 2: Use the formula for the determinant of a 3x3 matrix: det(I) = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign values from the matrix to the variables: a = 1, b = 0, c = 2, d = 0, e = 1, f = 3, g = 1, h = 0, i = 4.
Step 4: Calculate ei - fh: ei = 1*4 = 4 and fh = 3*0 = 0, so ei - fh = 4 - 0 = 4.
Step 5: Calculate di - fg: di = 0*4 = 0 and fg = 3*1 = 3, so di - fg = 0 - 3 = -3.
Step 6: Calculate dh - eg: dh = 0*0 = 0 and eg = 1*1 = 1, so dh - eg = 0 - 1 = -1.
Step 7: Substitute these values into the determinant formula: det(I) = 1(4) - 0 + 2(-1).
Step 8: Simplify the expression: det(I) = 4 - 0 - 2 = 2.

Determinants – The determinant of a matrix is a scalar value that can be computed from its elements and provides important properties about the matrix, such as whether it is invertible.
Cofactor Expansion – Cofactor expansion is a method for calculating the determinant of a matrix by breaking it down into smaller matrices, using minors and cofactors.