For each of the pairs of matrices that follow, determine whether it is possible to multiply the first matrix times the second. If it is possible, perform the multiplication. begin{bmatrix}1 & 4&3 0 & 1&40&0&2 end{bmatrix}begin{bmatrix}3 & 2 1 & 14&5 end{bmatrix}

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

If T :${\mathbb{R}}^2\to {\mathbb{R}}^2$ is a linear transformation such that
$T\left(\left[\begin{array}{c}3\\ 6\end{array}\right]\right)=\left[\begin{array}{c}39\\ 33\end{array}\right]\text{ and }T\left(\left[\begin{array}{c}6\\ -5\end{array}\right]\right)=\left[\begin{array}{c}27\\ -53\end{array}\right]$
then the standard matrix of T is A

find and sketch the domain of the function f(x,y)= x/x^2+y^2

Consider the matrices
$A=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}2& 3\\ 1& 5\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 8\end{array}\right],D=\left[\begin{array}{ccc}2& 0& -1\\ 1& 4& 3\\ 5& 4& 2\end{array}\right]\text{ and }F=\left[\begin{array}{ccc}2& -1& 0\\ 0& 1& 1\\ 2& 0& 3\end{array}\right]$
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $A{X}^T=B{C}^3$
(ii) $A^{-1}(X-T{)}^T=(B^{-1}{)}^T$
(iii) $XF=F^{-1}-D^T$

Find if possible the matrices:
a) AB
b) BA
$A=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$

 

If the roots of $a{x}^2+bx+c=0$ are $\frac{3}{2},\frac{4}{3}$ then $(a+b+c{)}^2=$