Let A be a square matrix of size n n (n>1). The elements of A=\a ij\ are given by a ij=casesi j,&i j\\0,&i<jcases. The determinant of A is

GATE 2021 CH – Question 36

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

Let A be a square matrix of size $n\times n$ $(n>1)$. The elements of $A=\{a_{ij}\}$ are given by $a_{ij}=\begin{cases}i\times j,&i\ge j\\0,&i<j\end{cases}$. The determinant of A is

0
1
$n!$
$(n!)^2$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $(n!)^2$

Explanation

Every entry above the main diagonal is zero, so A is **lower triangular**. Its determinant is the product of the diagonal entries; entries below the diagonal do not affect that product.

On the diagonal $i=j$, so $a_{ii}=i^2$. Hence
$$\det A=\prod_{i=1}^n i^2=(1\times2\times\cdots\times n)^2=\mathbf{(n!)^2}.$$

**Answer: D.**