If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix…

If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(k^2\) is:

[JEE Main 2021, 17 Mar (Shift 2)]