The three rules that define the determinant
Slide 3 measured det A as an area, and for 2 by 2 the formula ad − bc did the work. For 3 by 3 and bigger, we start from three rules instead. Think of a board game: learn three rules, and every move in the game follows from them. Rule 1: the identity I moves nothing, so det I = 1. Rule 2: swapping two rows flips the sign, like turning a page face down. Rule 3: the determinant is linear in each row: make one row twice as long and the volume doubles, just as a box with one side doubled holds twice as much. (Slide 3 built the box from the columns, so this needs care: doubling a row of A doubles one entry in every column. The volume still doubles, as slide 5 will show with det Aᵀ = det A. For now, check it with ad − bc.) Everything else follows from these three, including the most useful fact of all: elimination steps never change det.
Lesson 6 · slide 4Three rules- Rule 1: det I = 1. The identity does not change any volume.
- Rule 2: swapping two rows changes the sign of det.
- Rule 3: det is linear in each row on its own: scale a single row by t and det scales by t, and a row that is a sum splits det into a sum.
- Consequences: two equal rows give det = 0; adding a multiple of one row to another row does not change det.
Rule 1: det I = 1
The identity leaves every vector where it is, so the unit square (or cube) keeps area 1.
Rule 2: a row swap flips the sign
Swap the two rows of a matrix and its determinant changes sign. The size of the area stays, but the plane is turned over.
Rule 3: linear in each row, one row at a time
Two parts. First, multiply one row by t and det is multiplied by t. (Doubling a row doubles one entry in every column, so the box changes in a less obvious way; slide 5 shows, with det Aᵀ = det A, that the volume still doubles. Here we check it with ad − bc.)
Second, if one row is a sum, det splits into two determinants, one for each part. The other rows stay the same:
Careful: one row at a time. Doubling the whole n by n matrix doubles all n rows, so det is multiplied by 2ⁿ. And det(A + B) is usually not det A + det B. With A = B = I (2 by 2): det(2I) = 2·2 = 4, but det A + det B = 1 + 1 = 2.
What follows from the rules
Two equal rows give det = 0. Swapping them changes nothing, but rule 2 says the sign flips. So det = −det, and that means det = 0.
Subtracting ℓ times row j from row i leaves det unchanged (ℓ is the multiplier from lesson 2). The new row i is (row i) − ℓ(row j). By rule 3, det splits into det A − ℓ × det(A with row i replaced by row j). That second matrix has row j twice, so its det is 0. What is left is det A.
This is the big one: elimination steps (lesson 2) don't change det. The proof above uses rule 3 on rows, and the ad − bc check confirms it. For a picture: as a move of the plane, an elimination matrix E is a shear, which slants shapes without changing their size, like pushing a deck of cards sideways. Since EA is E times each column of A, the box of A's columns is slanted by E and keeps its volume. (Slide 5's det Aᵀ = det A gives the same answer for rows.)
A row of zeros gives det = 0. Multiply that row by 0: nothing changes, but rule 3 says det is multiplied by 0.
A worked check: one step, then a swap
Here is a 3 by 3 matrix whose determinant is 2. (Slides 5 and 6 show two ways to get that number; take it on trust for now.) One elimination step, then one row swap:
Keep reading with the full course
Finish lesson 6 and you will be able to: see a matrix as a move of the plane, and build one from where two arrows land.
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Examples
Example 1 · A row swap
det 1234 = 4 − 6 = −2.
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