In order to calculate the determinant for a matrix, it must be a square matrix, with equal rows and columns. The easiest square matrix to find the determinant is one that is 2×2. See Kelley’s Algebra pg. 121-122. Below, we will use his problem 5 and put it in J. The matrix is cross multiplied:
c d = a times d minus c times b = the determinant. In J it’s as follows:
As you can see, we can calculate two different ways. You can use separate cross multiplication steps using the scan multiply (*/) verb.
Then, do the subtraction (_9 – _12).
Or, we can do it all in one step using parentheses (9 * _1) – (3*_4)
Then, you can also check your work with the determinant command or verb (DET =: – / . *)
These 2×2 matrices are simple enough to do in pencil and paper but, if your arithmetic skills are unreliable, you have several ways to check your work.
Next, let’s experiment with Kelley’s Problem 5 and turn it into an actual system of equations.
9x – 4y = 13
3x – 1y = 9
As in Helzer’s Matrix divide, we take the variables to the left of the equal signs, which in fact are the original 2×2 matrix and assign the first column to x and the second column to y in Matrix A. Then we use the assignments to the right of the equal sign as Vector B below. The Determinant remains unchanged because the variables in A remain the same.
First, we use Matrix Divide (% .) to solve the matrix for x,y (7.66667 or 7 2/3, 14). Then, we use the DET verb to find and confirm the determinant for A.
Note that 7.66667 = 7 2/3 but that the % division operator always gives a decimal amount. I’ll explain fractions and the (r) operator in a future post.
Here is another example of the same kind of problem taken from the Intmath.com site. This is Example 2 from their site on how to use determinants and Cramer’s rule to solve a system of equations:
x – 3y = 6
2x + 3y = 3
We will use J Matrix divide and DET instead:
If you check the answers at the link, you will see that the answers for x,y (3, -1) and the DET (9) are the same as with J.
Next math post, 3 x 3 determinants.