Determinant of a Triangular Matrix - Linear Algebra Step by Step

Compute the determinant of an upper-triangular 3x3 matrix. Because all entries below the diagonal are zero, every cofactor term that involves a below-diagonal entry vanishes. Only the main diagonal product survives. The state panel records all three diagonal entries then multiplies them to produce the determinant.

Example

Read the diagonal entries and multiply them.

highlighted = computed this step

Step 1 — Set up

Start with the given matrix data.

A=\begin{bmatrix}2&3&1\\0&4&-2\\0&0&5\end{bmatrix}

Step 2 — Diagonal entry 1

Record d1 = 2.

d1=\hl{2}

Step 3 — Diagonal entry 2

Record d2 = 4.

d2=\hl{4}

Step 4 — Diagonal entry 3

Record d3 = 5.

d3=\hl{5}

Step 5 — Result

Multiply the diagonal entries: 2 * 4 * 5 = 40.

2\cdot 4\cdot 5=\hl{40}

triangular determinant For any triangular matrix (upper or lower), det(A) = product of the diagonal entries. The cofactor expansion collapses to a single term per row because every off-diagonal submatrix has a zero row or column.