Arrange the grid with two columns and two rows per column. If you label the entries consecutively as a, b, c and d, the matrix would look like:
a b
c d
Construct the formula by using the two-by-two matrix pattern. Essentially, you are subtracting the product of the diagonals:
Determinant = ad - cb
Remember when variables are placed side-by-side, they are multiplied, so the formula can also be interpreted as:
a * d - c * b
Solve for the determinant by plugging in your numbers. Suppose the variables a, b, c and d were 1, 2, 3 and 4, respectively:
Determinant = (1 x 4) - (2 x 3)
Determinant = 4 - 6
Determinant = -2
Arrange the grid with three columns and three rows per column. If you labeled the entries consecutively as a, b, c, d, e, f, g, h and i, the matrix would look like:
a b c
d e f
g h i
Construct the formula using the three-by-three matrix pattern:
Determinant = a(ei - hf) - d(bi - hc) + g(bf - ef)
Remember that variables placed side-by-side are multiplied. Therefore, this formula could also be expressed as:
[a x ((e x i) - (h x f))] - [d x ((b x i) - (h x c))] + [g x ((b x f) - (e x c))]
Although not immediately obvious, you are still multiplying diagonals, but there are three sections, each multiplied by the first entry of the corresponding row that is not included in the diagonals. In the first section, a is multiplied by the product of ei and hf, both diagonals and neither on the same row as a. Likewise, d is multiplied by the product of bi and hc, and g is multiplied by the product of bf and ec. The three sections are either subtracted or added together, depending on position.
Solve for the determinant by plugging in your numbers. Suppose the variables a, b, c, d, e, f, g, h and i are 1, 2, 3, 4, 5, 6, 7, 8 and 9, respectively:
Determinant = [1 x ((5 x 9) - (8 x 6))] - [4 x ((2 x 9) - (8 x 3))] + [7 x ((2 x 6) - (5 x 3))]
Determinant = [1 x (45 - 48)] - [4 x (18 - 24)] + [7 x (12 - 15)]
Determinant = [1 x -3] - [4 x -6] + [7 x -3]
Determinant = [-3 - (-24) + (-21)]
Determinant = 0