If $A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$,$A^{-1} = \alpha A + \beta I$ and $\alpha + \beta = -2$,then $4\alpha^2 + \beta^2 + \lambda^2$ is equal to:

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$,and $A^{-1} = \frac{1}{6}[A^2 + cA + dI]$ where $c, d \in R$,then the pair of values $(c, d)$ is:

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A \cdot \operatorname{adj} A = A^T$,then $5a + b$ is equal to

If $A$ is a $3 \times 3$ matrix and $|A|=\frac{1}{2}$, then $|A^{-1}(\operatorname{Adj}(\operatorname{Adj} A))|^{-1} = $

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A (adj A) = AA^T$, then $5a + b =$

If $A, B, C$ are three square matrices such that $AB = AC$ implies $B = C$,then the matrix $A$ is always a/an