\text { Consider the following vectors } \mathbf{a}^{T}=\left[\mathrm{ID}_{2} ; \quad \mathrm{ID}_{1} ; \mathrm{ID}_{3}\right], \mathbf{b}^{T}=\left[-\mathrm{ID}_{4} ; \mathrm{ID}_{5} ; \quad \mathrm{ID}_{6}\right], \mathbf{c}^{T}= \left[-\mathrm{ID}_{2} ; \quad 2 \mathrm{ID}_{3} ; \mathrm{ID}_{2}\right], \text { and

} \mathbf{d}^{T}=\left[\mathrm{ID}_{1} ; \quad \mathrm{ID}_{4} ; 2 \mathrm{ID}_{5}\right] . \text { Furthermore, consider the matrix } A=\left[\begin{array}{ccc} -\mathrm{ID}_{4} & -\mathrm{ID}_{2} & \mathrm{ID}_{2} \\ \mathrm{ID}_{5} & 2 \mathrm{ID}_{3} & \mathrm{ID}_{1} \\ \mathrm{ID}_{6} & \mathrm{ID}_{2} & \mathrm{ID}_{3} \end{array}\right] Fill in the corresponding values of the vectors and tables, for an easier progress: Find the rank of matrix A Are the vectors a, b, and c linearly independent? O Are the vectors b and c orthogonal? Find the dot product of the vectors a and c" Find the inverse of matrix A using the Analytical solution ) Find the solution vector for a linear system of equations that has A as a coefficient matrix and d as a solution vector (show your working).[5]