The three row operations of elimination change a determinant in three simple ways. Swapping two rows flips its sign, scaling a row scales it, and adding a multiple of one row to another leaves it alone. So a determinant can be found by reducing the matrix to a triangle.

Cutting a triangle off one end of a field and gluing it to the other keeps the area, and that is the third row operation. Measuring the field the other way round changes the sign, and stretching one side stretches the area.

The row operations are written $R_i\leftrightarrow R_j$ for exchanging rows $i$ and $j$, $R_i\to cR_i$ for multiplying row $i$ by $c$, and $R_i\to R_i+cR_j$ for adding $c$ times row $j$ to row $i$. They are the moves of elimination, made on the rows of a matrix instead of on equations. Each changes the determinant in a way fixed in advance. That a swap reverses the sign of every term is a fact about rearrangements — exchanging two numbers in one changes its count of inversions by an odd number — and it is used here without proof.

• Two equal rows give $\det A=0$: exchanging them changes nothing, yet changes the sign, and the only number equal to its own negative is $0$.
• A row that is a multiple of another gives $\det A=0$, and so does a row of zeros.
• The determinant is linear in each row on its own: if row $i$ is , the determinant is the sum of the two determinants with row replaced by and by . Each term contains exactly one entry of row , which is why.

Every term of the determinant contains exactly one entry of row $i$, so the determinant is linear in that row. Replacing row $i$ by row $i$ plus $c$ times row $j$ therefore splits it into the old determinant plus $c$ times the determinant of a matrix in which rows and are equal, and that second determinant is .

A term takes exactly one entry from every row, so it is an entry of row $i$ times a product of entries from the other rows.

Split every term along the sum in row $i$: the determinant is linear in that row while the other rows stay fixed.

The second matrix has row $j$ in two places. Exchanging those two rows changes nothing and yet flips the sign, so its determinant is $0$.

What remains is the determinant of the matrix before the operation.

• $R_i\leftrightarrow R_j$: the determinant changes sign.
• $R_i\to cR_i$: the determinant is multiplied by .

An arrow labelled $R_3-2R_1$ means row 3 is replaced by row 3 minus twice row 1. Such steps keep the determinant, a swap flips its sign, and at the end the diagonal of the triangle is multiplied.

The triangle has determinant $1\cdot2\cdot2=4$, and one swap was made, so the original determinant is $-4$.

• An exchange of two rows reverses the sign.
• Multiplying one row by $c$ multiplies the determinant by $c$; $\det(cA)=c^n\det A$.
• Adding a multiple of one row to another changes nothing.
• Equal rows, proportional rows or a zero row give determinant .

Great work! You now know how to:

• say what each row operation does to a determinant;
• compute a determinant by reducing to a triangle;
• use $\det(cA)=c^n\det A$ and the rule for a row that is a sum;
• spot determinants that must be zero.