List of top Mathematics Questions on Matrix Operations

If $f: \mathbb{N} \to \mathbb{Z}$ is defined by \[ f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}, k \in \mathbb{N}, \] and $\sum_{n=1}^k f(n) = 98$, then $k$ is equal to :

Let \( A = \begin{bmatrix} 1 & 0 & 0\\ 3 & 1 & 0\\ 9 & 3 & 1 \end{bmatrix} \) and \( B = [b_{ij}], 1 \le i,j \le 3 \). If \( B = A^{99} - I \), then the value of \( \dfrac{b_{31}-b_{21}}{b_{32}} \) is:

Let $ A = [a_{ij}] $ be a $3 \times 3 $ matrix such that: $$ A = \begin{bmatrix} 0 & 0 & 4 \\ 1 & 0 & 0 \\ 0 & 1 & 3 \\ \end{bmatrix}, A^{-1} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 0 \\ 2 & 1 & 0 \end{bmatrix}. $$ Then $ a_{23} $ equals: