# 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? Question
Matrices 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? 2021-02-10
Step 1
The given set is $$\left\{ \left. \begin{bmatrix}a & 1 \\ 1 & b \end{bmatrix} \right| a,b \in K\right\}$$
It is known that any subspace should contain the identity element.
Step 2
Note that the identity matrix $$\begin{bmatrix}1 & 0 \\0& 1 \end{bmatrix}$$ does not belongs to $$\left\{ \left. \begin{bmatrix}a & 1 \\ 1 & b \end{bmatrix} \right| a,b \in K\right\}$$ Thus, the answer is NO. the set of all 2 x 2 matrices of the form $$\left\{ \left. \begin{bmatrix}a & 1 \\ 1 & b \end{bmatrix} \right| a,b \in K\right\}$$ is not a subspace of all 2 x 2 matrices.

### Relevant Questions 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 $$U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}$$ 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. Let W be the subspace of all diagonal matrices in $$M_{2,2}$$. Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix 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? Determine whether the subset of $$M_{n,n}$$ is a subspace of $$M_{n,n}$$ with the standard operations. Justify your answer.
The set of all $$n \times n$$ matrices whose entries sum to zero Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
and define $$M^{#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}$$ Characterize the matrices M such that $$M^{#}=M^{-1}$$ Let W be the vector space of $$3 \times 3$$ symmetric matrices , $$A \in W$$ Then , which of the following is true ?
a) $$A^T=1$$
b) $$dimW=6$$
c) $$A^{-1}=A$$
d) $$A^{-1}=A^T$$ If 4A-3B=2C (where A,B and C are all matrices) then Matrix A can be defined as:
Select one:
a) 0.5C+3B
b) $$\frac{2C+3B}{4}$$
c) 0.5C+0.75B
d) C+B If $$A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}$$
then AB=?
BA=?
True or false : AB=BA for any two square matrices A and B of the same size. Let B be a $$4 \times 4$$ matrix to which we apply the following operations: