When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve: x + y + z = 4, x + 2y + 2z = 2, 2x + 2y + z = 5.

What is the condition of a matrix to have an inverse? Find the inverse of the matrix \(A = \begin{bmatrix} 5 & 1 & 2\\ 1 & 0 & 3\\ 4 & -3 & 8 \end{bmatrix}\)  If it exists.

Define linearly independent set of vectors with an example. Show that the vectors (1, 4, 3), (0, 3, 1) and (3, -5, 4) are linearly independent. Do they form a basis? Justify.

Find the least-square solution of Ax = b for A = \(\begin{bmatrix}1 & 3 & 5\\ 1 & 1 & 0\\ 1 & 1 & 2\\ 1 & 3 & 3\end{bmatrix}\) and b= \(\begin{pmatrix}3\\ 5\\ 7\\ 3\end{pmatrix}\)

Change into reduce echelon form of the matrix \(\begin{pmatrix}0 & 3 & -6\\ 3 & -7 & 8\\ 3 & -9 & 12\end{pmatrix}\).

Define linear transformation with an example. Is a transformation defined by T(x, y) = (3x + y, 5x + 7y, x + 3y) linear? Justify.

Let \(A = \begin{pmatrix}-1 & -2\\ 5 & 9\end{pmatrix}\) and \(b = \begin{pmatrix}9 & 2\\ k & -1\end{pmatrix}\). What value (s) of k if any will make AB = BA?

Define determinant. Evaluate without expanding \(\begin{vmatrix}1 & 5 & -6\\ -1 & -4 & 4\\ -2 & -7 & 9\end{vmatrix}\)

Define subspace of a vector space. Let \(H = \left \{ \begin{pmatrix}s\\ t\\ 0\end{pmatrix}:s,t \in R \right \}\). Show that H is a subspace of:

Find the dimension of the null space and column space of \(A = \begin{bmatrix}-3 & 6 & -1 & 1 & -7\\ 1 & -2 & 2 & 3 & -1\\ 2 & -4 & 5 & 8 & -4\end{bmatrix}\)

Find the eigenvalues of the matrix \(\begin{pmatrix}6 & 3 & -8\\ 0 & -2 & 0\\ 1 & 0 & -3\end{pmatrix}\)

Find LU factorization of the matrix \begin{pmatrix}2 & 5\\ 6 & -7\end{pmatrix}

Define group. Show that the set of all integers Z forms group under addition operation.

Define ring with an example. Compute the product in the given ring (-3, 5) (2, -4) in Z₄ x Z₁₁.

State and prove the Pythagorean theorem of two vectors and verify this for u = (1, -1) and v = (1, 1).