# Write the set in the form {x|P(x)}, where P(x) is a property that described the elements of the set. {a,e,i,o,u}. Question
Matrices Write the set in the form {x|P(x)}, where P(x) is a property that described the elements of the set. {a,e,i,o,u}. 2021-02-25
Notice that all elements of the set are letters. Moreover, they are vowels. Thus
{x|x is a vowel}

### Relevant Questions In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix. Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample. 1. Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector subspace of all 2 x 2 matrices? Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$. $$c_1\cdot\begin{cases}x_1+x_2+x_3=0\\5x_1-2x_2+2x_3=0\\8x_1+x_2+5x_3=0\end{cases},\ X=c_1\begin{pmatrix}4\\3\\-7\end{pmatrix}$$ Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set:
$$\displaystyle{P}{\left({A}\right)}={\left\lbrace{x}{\mid}{x}\subseteq{A}\right\rbrace}$$
Show that symmetric deference operation on P(A) define by the formula
$$\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)},{x}\in{P}{\left({A}\right)},{y}\in{p}{\left({A}\right)}$$
(where $$\displaystyle{y}^{{c}}$$ is the complement of y) the following statement istrue:
The algebraic operation o+ is commutative and associative on P(A). Let $$M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})$$ be the set of 2 x 2 matrices with the entries in $$\mathbb{Z}/\mathbb{6Z}$$
a) Can you find a matrix $$M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})$$ whose determinant is non-zero and yet is not invertible?
b) Does the set of invertible matrices in $$M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})$$ form a group? (square roots of the identity matrix) For how many 2x2 matrices A is it true that $$A^2=I$$ ? Now answer the same question for n x n matrices where n>2 Let V be the vector space of real 2 x 2 matrices with inner product
(A|B) = tr(B^tA).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for $$U^\perp$$ where $$U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}$$ $$A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}$$ Let $$u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}$$ It can be shown that 4u-3v-w=0. Use this fact (and no row operations) to find a solution to the system Ax=b , where
$$A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}$$