It can be shown that the algebraic multiplicity of an eigenvalue lambda is always greater than or equal to the dimension of the eigenspace correspondi

Assume that A is row equivalent to B. Find bases for Nul A and Col A.
$A=\left[\begin{array}{ccccc}1& 2& -5& 11& -3\\ 2& 4& -5& 15& 2\\ 1& 2& 0& 4& 5\\ 3& 6& -5& 19& -2\end{array}\right]$
$B=\left[\begin{array}{ccccc}1& 2& 0& 4& 5\\ 0& 0& 5& -7& 8\\ 0& 0& 0& 0& -9\\ 0& 0& 0& 0& 0\end{array}\right]$

Find an explicit description of Nul A by listing vectors that span the null space.
$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$
Find a nonzero vector in Nul A.
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Solve the given system of equations, or else show that there is no solution.
a) ${\displaystyle x_1+2x_2-x_3=1}$
${\displaystyle 2x_1+x_2+x_3=1}$
${\displaystyle x_1-x_2+2x_3=1}$
b) ${\displaystyle x_1+2x_2-x_3=-2}$
${\displaystyle -2x_1-4x_2+2x_3=4}$
${\displaystyle 2x_1+4x_2-2x_3=-4}$

Write an augumented matrix for the system of linear equations
${\displaystyle \left[\begin{array}{c}x-y+z=8\\ y-12z=-15\\ z=1\end{array}\right]}$

Unimodular Matrix is a matrix, which has integer entries and determinant of $\in \{+1,-1\}$. That means the matrix is invertible over the integers $\mathbb{Z}$.
Given a unimodular matrix A for example:
$\left[\begin{array}{cccccc}2& -2& -5& -4& -2& -3\\ 0& 1& 0& 0& 0& 0\\ -1& 0& 3& 1& -1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
Need to convert to it to a matrix like this:
$\left[\begin{array}{cccccc}\ast & \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & 0& 0& 0& 0\\ \ast & 0& \ast & 0& 0& 0\\ \ast & 0& 0& \ast & 0& 0\\ \ast & 0& 0& 0& \ast & 0\\ \ast & 0& 0& 0& 0& \ast \end{array}\right]$
where * denotes possible nonzero entries.

Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is ${\displaystyle {\mathrm{\forall }}^T}$ something special for any matrix A?