Cramer's Rule, Explained Geometrically

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Cramer's rule is usually taught as a formula to memorize, with no reason given. That is a shame, because the geometric story behind it is clear and even pretty. In practice you will often solve systems other ways, but understanding Cramer's rule cements your feel for determinants.

The problem we are solving

We are back to solving A * x = v, the same setup as Chapter 7. For a 2D example:

| a  b |   | x |   | e |
| c  d | * | y | = | f |

We want the unknowns x and y. Cramer's rule gives them directly as ratios of determinants. The question is why that works.

A first idea that almost works

You might guess that x is somehow connected to the coordinates of the output v. But a transformation scrambles coordinates, so you cannot just read x off v. We need a quantity that survives the transformation in a predictable way. That quantity is area.

The key picture: area tied to a coordinate

Here is the trick. In the input space, build a parallelogram using the first basis vector î and the unknown solution vector [x, y].

The area of that parallelogram (signed) turns out to be exactly y. Why? Because î has length 1 along the x-axis, so the parallelogram's area is just its height, which is the y coordinate of the solution. Similarly, the parallelogram made from [x, y] and ĵ has signed area equal to x.

So each unknown coordinate is hiding as the area of a simple parallelogram.

Now apply the transformation

When we run the transformation A, every area gets scaled by det(A) (that was the whole point of Chapter 6). So:

• The parallelogram that had area x now has area x * det(A).
• But after the transformation, that same parallelogram is built from the transformed pieces: the output vector v and the second column of A (which is where ĵ landed).

So the area of that transformed parallelogram can be computed two ways. One way gives x * det(A). The other way is a determinant of a matrix made from v and the columns of A.

Set them equal and solve for x. That is Cramer's rule.

The formula

For the 2D system above:

x = det | e  b |  divided by  det | a  b |
        | f  d |                  | c  d |

y = det | a  e |  divided by  det | a  b |
        | c  f |                  | c  d |

Read the pattern. The bottom is always det(A). For x, you replace the first column of A with the output vector v, then take its determinant. For y, you replace the second column with v. Each unknown gets its own column swapped in.

A quick number example:

2x + 1y = 5
1x + 3y = 6

det(A) = det | 2  1 | = (2)(3) - (1)(1) = 5
             | 1  3 |

x = det | 5  1 |  / 5 = (5*3 - 1*6) / 5 = (15 - 6) / 5 = 9/5
        | 6  3 |

y = det | 2  5 |  / 5 = (2*6 - 5*1) / 5 = (12 - 5) / 5 = 7/5
        | 1  6 |

So x = 9/5 and y = 7/5. You can plug them back in to check; they fit both equations.

When it breaks

Notice the formula divides by det(A). If det(A) is zero, you cannot divide, and Cramer's rule fails. That is the same condition from Chapter 7: a zero determinant means space got squished, the transformation is not reversible, and there is no single clean solution. Everything stays consistent.

Should you actually use it?

Cramer's rule is elegant and great for understanding, and it is handy for tiny 2 by 2 or 3 by 3 systems by hand. But for large systems it is slow compared to other methods like elimination, because computing many determinants is expensive. So treat this chapter as intuition-building first, and a practical tool only for small cases.

Quick gotchas

Swap the right column for each unknown. First column for x, second for y, and so on. The output vector v goes into the column matching the variable you want.

Always divide by det(A). If it is zero, the rule does not apply.

It is slow for big systems. Beautiful for understanding, not the fast tool for large problems.

What you walked away with

• Each unknown coordinate equals the signed area of a small parallelogram.
• The transformation scales all areas by det(A), so dividing undoes that scaling.
• Cramer's rule: replace one column of A with the output v, take that determinant, divide by det(A).
• It fails exactly when det(A) = 0, matching everything from Chapter 7.

Next up, Chapter 13: change of basis. We will learn to translate vectors and transformations between different coordinate systems, the formal version of "the same arrow gets different numbers in a different basis".