The determinant of an upper triangular matrix is the x of its diagonal elements.

Solution

Maths sir · 3 steps

Expand down the first column: only the top-left entry survives, every time, all the way down.

1. Look at the zeros. In an upper triangular matrix, every entry below the main diagonal is 0. So the first column has only one possibly non-zero entry, a₁₁, at the top.

   |a₁₁, a₁₂, a₁₃; 0, a₂₂, a₂₃; 0, 0, a₃₃| = a₁₁|a₂₂, a₂₃; 0, a₃₃| = a₁₁a₂₂a₃₃

2. Repeat the expansion. Expanding along the first column leaves a₁₁ times a smaller determinant that is again upper triangular. Doing this again and again peels off one diagonal entry each time.
3. So. The determinant is the product of the diagonal elements, a₁₁a₂₂… aₙₙ. The answer key records this answer as 1.

The idea to keep. For any triangular matrix, upper or lower, the determinant is the product of the diagonal entries. If even one diagonal entry is zero, the determinant is zero.

Answer from the NTA final answer key. Solution by PrepWiser.