Question

A square matrix AA is called half-magic if the sum of the numbers in each row and column is the same. The common sum in each row and column is denoted by s(A)s(A) and is called the magic sum of the matrix AA. Let VV be the vector space of 2×22×2 half-magic squares.
A) Find an ordered basis BB for VV.
B) Find the coordinate vector [M]_B of M [-2 -7, -7 -2]

Answer 1

Answer:

A)  $B = \{\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right], \left[\begin{array}{ccc}0&1\\1&0 \end{array}\right] \}$

B) $M_{B} = \left[\begin{array}{ccc}-2\\-7\end{array}\right]$

Step-by-step explanation:

Let $A = \left[\begin{array}{ccc}a&b\\c&d \end{array}\right]$ 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  $A = \left[\begin{array}{ccc}a&b\\b&a \end{array}\right]$ where $a,b \epsilon R$

A can therefore be manipulated as:

$A = a \left[\begin{array}{ccc}1&0\\0&1 \end{array}\right] + b \left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]$

The matrices $\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right]$ and $\left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]$ are apparently linearly independent and therefore form a basis B for V

$B = \{\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right], \left[\begin{array}{ccc}0&1\\1&0 \end{array}\right] \}$

B) Find the coordinate vector [M]_B of M [-2 -7, -7 -2]

 $M = \left[\begin{array}{ccc}-2&-7\\-7&-2 \end{array}\right]$

$M$ can be written in the form $M = a\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right] + b\left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]$

$M = \left[\begin{array}{ccc}-2&-7\\-7&-2 \end{array}\right] = -2\left[\begin{array}{ccc}1&0\\0&1 \end{array}\right] -7\left[\begin{array}{ccc}0&1\\1&0 \end{array}\right]$

The coordinate vector is therefore, $M_{B} = \left[\begin{array}{ccc}-2\\-7\end{array}\right]$

Answer 2

Answer:

A square matrix AA is called half-magic if the sum of the numbers in each row and column is the same

a) Find an ordered basis B for V.

A 2x2 half-magic matrix has the form:

$A = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right]$

Where a, b, c, and d are real numbers.

Hence,

a+b = a+c = b+d = c+d

Where, b=c, & a=d

Therefore,

$A = \left[\begin{array}{ccc}a&b\\b&a\end{array}\right] = a \left[\begin{array}{ccc}1&0\\0&1\end{array}\right] + b \left[\begin{array}{ccc}0&1\\1&0\end{array}\right]$

The matrices $\left[\begin{array}{ccc}1&0\\0&1\end{array}\right] and \left[\begin{array}{ccc}0&1\\1&0\end{array}\right]$ are linearly independent.

An ordered basis B for V is:

$[\left[\begin{array}{ccc}1&0\\0&1\end{array}\right], \left[\begin{array}{ccc}0&1\\1&0\end{array}\right]]$

b) Find the coordinate vector $[M]_B of M = \left[\begin{array}{ccc}-2&-7\\-7&-2\end{array}\right]$

Here,

$M = \left[\begin{array}{ccc}-2&-7\\-7&-2\end{array}\right] = -2 \left[\begin{array}{ccc}1&0\\0&1\end{array}\right] = -7 \left[\begin{array}{ccc}0&1\\1&0\end{array}\right]$

Hence,

$M_B = \left[\begin{array}{ccc}-2\\-7\end{array}\right]$