Matrix Transformation Examples and Practice Problems

Find k such that the following matrix M is singular.
${\displaystyle \left[\begin{array}{ccc}-3& -3& 3\\ 6& 7& -6\\ 11+k& 16& -15\end{array}\right]}$

Is division of matrices possible?
Is it possible to divide a matrix by another? If yes, What will be the result of $\frac{A}{B}$ if
$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

$B=\left(\begin{array}{cc}w& x\\ y& z\end{array}\right)?$

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?

Let ${\displaystyle a_1=\left[\begin{array}{c}1\\ 3\\ -1\end{array}\right],a_2=\left[\begin{array}{c}-6\\ -14\\ 3\end{array}\right],b=\left[\begin{array}{c}5\\ 3\\ h\end{array}\right]}$. for what value(s) of h is b in the plane spanned by ${\displaystyle a_1}$ and ${\displaystyle a_2}$

Find the characteristic equation and the eigenvalues (and corresponding eigenvectors) of the matrix.
$$\left[\begin{array}{ccc}2& -2& 3\\ 0& 3& -2\\ 0& -1& 2\end{array}\right]$$
a) The characteristic equation
${\displaystyle {\lambda }^3-7{\lambda }^2+14\lambda -8=0}$
b) The eigenvalues
${\displaystyle ({\lambda }_1,{\lambda }_2,{\lambda }_3)=(1,2,4)}$
The corresponding eigenvectors
${\displaystyle x_1=-7,1,1?}$
${\displaystyle x_2=1,0,0}$
${\displaystyle x_3=13,-4,2?}$

Detemine if ${\displaystyle b}$ is a linear combination of ${\displaystyle a_1,a_2,a_3}$
$$a_1=\left[\begin{array}{c}2\\ 0\\ 2\end{array}\right],a_2=\left[\begin{array}{c}-4\\ 3\\ -4\end{array}\right],a_3=\left[\begin{array}{c}-5\\ 8\\ 4\end{array}\right],b=\left[\begin{array}{c}13\\ -4\\ 9\end{array}\right]$$
Choose the correct answer below
A. Vector ${\displaystyle b}$ is a linear combination of ${\displaystyle a_1,a_2,a_3}$. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
B. Vector ${\displaystyle b}$ is not a lincar combination of ${\displaystyle a_1,a_2,a_3}$
C. Vector ${\displaystyle b}$ is a linear combination of ${\displaystyle a_1,a_2,a_3}$. The pivots in the corresponding echelon matrix are in the first entry in the first column and the third entry in the second column, and the third entry in the third column.
D. Vector ${\displaystyle b}$ is a linear combination of ${\displaystyle a_1,a_2,a_3}$. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column.

Assume that T is a linear transformation. Find the standard matrix of T. ${\displaystyle T:{\mathbb{R}}^2\Rightarrow {\mathbb{R}}^2}$ rotates points (about the origin) through - ${\displaystyle \frac{\pi }{4}}$ radians (clockwiese).${\displaystyle [H\int :T\left(e_1\right)=(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}).]}$

Use an inverse matrix to solve each system of linear equations.
2x-y=-1
2x+y=-3

Find the value(s) of kk such that AA is singular.
$$\left[\begin{array}{cc}k-1& 2\\ 2& k+2\end{array}\right]$$

Construct a matrix whose column space contains (1, 1, 5) and (0, 3.1) and whose nullspace contains (1, 1, 2).

Use an inverse matrix to solve each system of linear equations.
2x-y=-3
2x+y=7

Explain why the columns of an n x n matrix A are linearly independent when A is invertible.

Let A be an invertible ${\displaystyle n\times n}$ matrix, and let B be an ${\displaystyle n\times p}$ matrix. Show that the equation AX=B has a unique solution ${\displaystyle A^{-1}B}$.

If we have a matrices
$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{ and }{e}_{12}(\lambda )=\left[\begin{array}{cc}1& \lambda \\ 0& 1\end{array}\right]$$
then by doing product $$A{e}_{12}(\lambda )=\left[\begin{array}{cc}a& a\lambda +b\\ c& c\lambda +d\end{array}\right]\text{ and }{e}_{12}(\lambda )A=\left[\begin{array}{cc}a+c\lambda & b+d\lambda \\ c& d\end{array}\right]$$
we can interpret that right multiplication by ${\displaystyle e_{12}}$ to A gives a column-operation: add ${\displaystyle \lambda }$-times first column to the second column. In similar way, left multiplication by ${\displaystyle e_{12}\left(\lambda \right)}$ to A gives row-operation on A.
Is there any conceptual (not computational, if any) way to see that elementary row and column operations on a matrix can be expressed as multiplication by elementary matrices on left or right, accordingly?

Find the change-of-coordinates matrix from ${\displaystyle \mathcal{B}}$ to the standard basis in ${\displaystyle {\mathbb{R}}^n}$
$$\mathcal{B}=\{\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 0\\ -5\end{array}\right],\left[\begin{array}{c}8\\ -2\\ 7\end{array}\right]\}$$

A and B are ${\displaystyle n\times \cap \times n}$ matrices. Mark each statement True or False. Justify your answer.
a. A row replacement operation does not affect the determinant of a matrix.
b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by ${\displaystyle {(-1)}^r}$ , where r is the number of row interchanges made during row reduction from A to U
c. If the columns of A are linearly dependent, then det ${\displaystyle A=0.d.det(A+B)=detA+detB}$.

Write the system first as a vector equation and then as a matrix equation.
${\displaystyle 8x_1-x_2=4}$
${\displaystyle 5x_1+4x_2=1}$
${\displaystyle x_1-3x_2=2}$

Find all x in ${\displaystyle {\mathbb{R}}^4}$ that are mapped into the zero vector by the transformation ${\displaystyle \mathbf{X}\mapsto A\mathbf{X}}$ for the given matrix A.
$$A=\left[\begin{array}{cccc}1& 3& 9& 2\\ 1& 0& 3& -4\\ 0& 1& 2& 3\\ -2& 3& 0& 5\end{array}\right]$$

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