Sarrus Rule Calculator guide
Rule of Sarrus: A Complete Worked Example
For the matrix with rows (2, 2, −5), (3, 1, 5) and (7, 2, 6): the down-right products are 12, 70 and −30, totalling 52. The up-right products are −35, 20 and 36, totalling 21. The determinant is 52 − 21 = 31.
Why this particular matrix
This example was chosen to be slightly awkward on purpose. It has a negative entry in it, and none of the products come out as round numbers, so there is nowhere for a mistake to hide.
If you work through it once carefully alongside the calculator, comparing each of your six products against each of its six, the method tends to stay put. Rushing a clean example teaches you much less than finishing a messy one.
Widen the grid
Copy column 1 and column 2 to the right of the matrix, giving a grid that is three rows tall and five columns wide.
In the calculator on this page the two copied columns are drawn with dashed outlines, so you can always tell which numbers are your original matrix and which are the repeats.
The three down-right products
Starting from each entry of the top row, follow the diagonal down and to the right:
- 2 × 1 × 6 = 12
- 2 × 5 × 7 = 70
- −5 × 3 × 2 = −30
- Adding them: 12 + 70 − 30 = 52
The three up-right products
Now from each entry of the bottom row, follow the diagonal up and to the right:
- 7 × 1 × −5 = −35
- 2 × 5 × 2 = 20
- 6 × 3 × 2 = 36
- Adding them: −35 + 20 + 36 = 21
Subtract, and read the answer
det = 52 − 21 = 31.
Since 31 is not zero, this matrix is invertible and its three rows are linearly independent.
It is worth pausing on how that single negative entry travelled through the calculation. It made one of the down-right products negative and one of the up-right products negative, and those two ended up on opposite sides of the subtraction. Keeping track of signs is where most Sarrus errors happen, which is the whole argument for writing each product on its own line rather than trying to hold them in your head.
Try changing one number
A good way to build a feel for this is to change a single entry in the calculator and watch which of the six products move.
Change the top-left 2 to a 3, for instance, and only the two products that pass through that cell will change. That is a useful thing to have noticed the next time your answer disagrees with the back of the book: the wrong product is almost always one that passes through the cell you misread.
