Answer:
A) ![B = \{\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right], \left[\begin{array}{ccc}0&1\\1&0 \end{array}\right] \}](https://tex.z-dn.net/?f=B%20%3D%20%5C%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%20%5Cend%7Barray%7D%5Cright%5D%2C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%261%5C%5C1%260%20%5Cend%7Barray%7D%5Cright%5D%20%5C%7D)
B) ![M_{B} = \left[\begin{array}{ccc}-2\\-7\end{array}\right]](https://tex.z-dn.net/?f=M_%7BB%7D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-2%5C%5C-7%5Cend%7Barray%7D%5Cright%5D)
Step-by-step explanation:
Let
where a, b, c and d are real numbers
Since A is said to be a half magic square matrix, a = d, b = c.
The matrix A therefore becomes
where 
A can therefore be manipulated as:
![A = a \left[\begin{array}{ccc}1&0\\0&1 \end{array}\right] + b \left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]](https://tex.z-dn.net/?f=A%20%3D%20a%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%20%5Cend%7Barray%7D%5Cright%5D%20%2B%20b%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%261%5C%5C1%260%20%5Cend%7Barray%7D%5Cright%5D)
The matrices
and
are apparently linearly independent and therefore form a basis B for V
![B = \{\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right], \left[\begin{array}{ccc}0&1\\1&0 \end{array}\right] \}](https://tex.z-dn.net/?f=B%20%3D%20%5C%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%20%5Cend%7Barray%7D%5Cright%5D%2C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%261%5C%5C1%260%20%5Cend%7Barray%7D%5Cright%5D%20%5C%7D)
B) Find the coordinate vector [M]_B of M [-2 -7, -7 -2]
can be written in the form ![M = a\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right] + b\left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]](https://tex.z-dn.net/?f=M%20%3D%20a%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%20%5Cend%7Barray%7D%5Cright%5D%20%2B%20%20b%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%261%5C%5C1%260%20%5Cend%7Barray%7D%5Cright%5D)
![M = \left[\begin{array}{ccc}-2&-7\\-7&-2 \end{array}\right] = -2\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right] -7\left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]](https://tex.z-dn.net/?f=M%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-2%26-7%5C%5C-7%26-2%20%5Cend%7Barray%7D%5Cright%5D%20%3D%20-2%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%20%5Cend%7Barray%7D%5Cright%5D%20-7%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%261%5C%5C1%260%20%5Cend%7Barray%7D%5Cright%5D)
The coordinate vector is therefore, ![M_{B} = \left[\begin{array}{ccc}-2\\-7\end{array}\right]](https://tex.z-dn.net/?f=M_%7BB%7D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-2%5C%5C-7%5Cend%7Barray%7D%5Cright%5D)