Determinants & Their Properties

The last lesson left us wanting one number, computed once, that tells whether a square matrix’s columns are independent — whether Ax = b has its single unique answer. There is such a number, and it tells you more besides: whether this matrix can be undone, by how much it stretches space, and whether its rows are secretly saying the same thing. One number — the determinant — answers all of it.

What it measures — a scaling of area

Think of a 2×2 matrix as a transformation of the plane. It takes the unit square (the box spanned by (1,0) and (0,1)) and sends it to a parallelogram. The determinant is the area of that parallelogram — the factor by which the transformation scales area. A determinant of 3 triples areas; a determinant of 1 leaves them alone; a determinant of 0 collapses the square flat onto a line, so the area vanishes. In 3-D the same story holds with volume.

Drag the matrix entries below and watch the parallelogram breathe. Its signed area is the determinant — make the two columns line up and the area collapses to zero, exactly the singular case from last lesson (rank below full). A negative reading just means the transform flipped orientation, like a mirror.

How to compute it

For a 2×2 matrix the rule is one line — the main-diagonal product minus the off-diagonal product:

For a 3×3 matrix, use cofactor expansion along the first row: multiply each entry by the 2×2 determinant of the rows and columns that remain, alternating signs + − +:

det([[a, b, c],      | e  f |        | d  f |        | d  e |
     [d, e, f],  = a·| h  i |  −  b· | g  i |  +  c· | g  h |
     [g, h, i]])

  = a(ei − fh) − b(di − fg) + c(dh − eg)

You may expand along any row or column (keeping the sign pattern), so pick the one with the most zeros to save work.

The properties that get tested

A handful of facts answer most exam questions with almost no computation:

• Product rule: det(AB) = det(A)·det(B) — determinants turn a matrix product into ordinary multiplication.
• Transpose: det(Aᵀ) = det(A) — transposing never changes it.
• Powers: det(Aᵏ) = det(A)ᵏ, straight from the product rule.
• Scalar: for an n×n matrix, det(cA) = cⁿ·det(A) — the scalar comes out once per row, so it is raised to the nth power.
• The big equivalence: det(A) = 0 ⇔ A is singular (no inverse) ⇔ its rows/columns are dependent ⇔ 0 is an eigenvalue. Four ways to say the same thing: the transformation flattened space and lost a dimension, so nothing can undo it.

A worked example — the 2024 trap

Part A — a clean 2×2. Compute det([[2, 1], [3, 4]]):

det = (2)(4) − (1)(3) = 8 − 3 = 5

The parallelogram spanned by the columns has area 5, and since it is non-zero the matrix is invertible.

Part B — the 2024 style. A 3×3 matrix M has linearly dependent rows (say row 3 is row 1 plus row 2). Find det(M² + 12M). Do not multiply anything — factor M out first:

M² + 12M = M·(M + 12I)

det(M² + 12M) = det( M·(M + 12I) )
              = det(M) · det(M + 12I)      ← product rule

Dependent rows make M singular, so det(M) = 0, and the whole product is 0 — the entries of M never mattered:

det(M² + 12M) = 0 · det(M + 12I) = 0

Spotting the dependent rows (hence det(M) = 0) and factoring with the product rule is the entire solution.

A question to carry forward

When det(A) ≠ 0, the matrix scales space without flattening it, so its action can be undone — there is a matrix that reverses it. Here is the thread onward: what is that reversing matrix called, how do we actually build it, and why does the determinant sit right in the denominator of the formula?

In one breath

• The determinant is one number per square matrix: the signed area/volume scale factor of its transformation.
• Compute: 2×2 is ad − bc; 3×3 is cofactor expansion a(ei−fh) − b(di−fg) + c(dh−eg) (expand along the row/column with the most zeros).
• Properties: det(AB)=det(A)det(B), det(Aᵀ)=det(A), det(Aᵏ)=det(A)ᵏ, det(cA)=cⁿdet(A). No sum rule — det(A+B) ≠ det(A)+det(B).
• The equivalence: det(A)=0 ⇔ singular ⇔ rows/cols dependent ⇔ 0 is an eigenvalue (space flattened, no inverse).
• The exam move: factor the determinant out with the product rule; never multiply the matrices.

Practice