For the given systems of linear equations, determine the values of b_1, b_2, text{ and } b_3 necessary for the system to be consistent. (Using matrices) x-y+3z=b_1 3x-3y+9z=b_2 -2x+2y-6z=b_3

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.

A poll is given, showing 35% are in favor of a new building project.If 3 people are chosen at random, what is the probability that exactly 1 of them favor the new building project?

Let V be the vector space of real 2 x 2 matrices with inner product
$(A|B)=tr(B^{t}A)$.
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for $U^{\perp }$ where $U^{\perp }\{A\in V|(A|B)=0\mathrm{\forall }B\in U\}$

Sketch the graph of the inequality.(x + 2)2 + y2 > 1

Solve the system of equations using matrices.Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\{\begin{array}{l}x+y-z=-2\\ 2x-y+z=5\\ -x+2y+2z=1\end{array}$

Find the inverse of the matrix using elementary matrices. $[[2,0],[1,1]]$