List of top Mathematics Questions on Matrices asked in NATA

Let \( a, b, c \) be distinct real numbers such that
\[\Delta = \begin{vmatrix}2014 & 2015 & 2013 + a^{-1} \\2015 & 2016 & 2013 + b^{-1} \\2016 & 2017 & 2013 + c^{-1}\end{vmatrix} = 0\]
Then:

Value of the determinant \[\begin{vmatrix}1 & w & w^2 \\w & w^2 & 1 \\w^2 & 1 & w\end{vmatrix}\] where \( w \) is the cube root of unity is:

If \( A = \begin{bmatrix} -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{bmatrix} \), then \( A^{-1} \) is: