Q.Find the value of the following: $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$

The key idea is that this is an upper triangular matrix — all entries below the main diagonal are zero. For any triangular matrix, the determinant is simply the product of the diagonal entries.

Step 1: Identify the diagonal elements: $1$, $2$, and $5$.

Step 2: Multiply them: $1 \times 2 \times 5 = 10$. …

The problem asks for the determinant of a $3 \times 3$ matrix. Before diving into expansion, notice the structure: all entries below the main diagonal are zero. That’s the hallmark of an upper triangular matrix.

Why triangular matrices are special

For any square matrix that is triangular (upper or lower), the determinant equals the product of the diagonal elements. This isn’t a coincidence — it follows directly from the definition of the determinant using cofactor expansion.

When you expand along the first column of an upper triangular matrix, the only non-zero term is the top-left entry times the determinant of the smaller triangular submatrix. Repeating this process down the diagonal gives the product.

Step-by-step verification

Let’s confirm by expanding the given matrix:

1. Expand along the first column — it has only one non-zero entry, $1$, at position $(1,1)$. The cofactor is $(-1)^{1+1} = 1$ times the determinant of the $2 \times 2$ submatrix formed by deleting row 1 and column 1: $\det(A) = 1 \cdot \det\begin{bmatrix} 2 & 4 \\ 0 & 5 \end{bmatrix}.$ …

Method: Evaluating the Determinant of a Triangular Matrix

Use this whenever the matrix has all zeros on one side of the main diagonal.

Steps

Step 1: Confirm the matrix is upper (or lower) triangular

Check that every entry above (or below) the main diagonal is $0$.

Step 2: Read off the diagonal entries

Identify $a_{11}, a_{22}, a_{33}, \dots$ down the main diagonal.

Step 3: Multiply only the diagonal entries together …

Common Mistakes

Mistake 1: Multiplying ALL entries of the matrix instead of only the diagonal ones

Why it's wrong: the triangular shortcut only works because the off-diagonal zeros make every other cofactor-expansion term vanish — including the nonzero off-diagonal entries (like $2$ and $3$ here) in the product gives a completely wrong, inflated answer. Correct approach: identify the diagonal entries specifically ($1,2,5$ here) and multiply only those.

Mistake 2: Assuming the triangular shortcut applies without first confirming the matrix really is triangular …