Question(1 point)
A square matrix is half-magic if the sum of the numbers in each row and column is the same. Find a basis $B$ for the vector space of $2 \times 2$ half-magic $B=\left\{\left[\begin{array}{l} \square \\ \square \square \square \end{array}\right] \cdot\left[\begin{array}{l} \square \square \\ \square \\ \square \end{array}\right]\right\}$ Preview My Answers Submit Answers
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Studdy Solution

STEP 1

1. A $2 \times 2$ half-magic matrix has the property that the sum of the elements in each row is equal to the sum of the elements in each column.
2. We are looking for a basis for the vector space of such matrices, meaning we need to find a set of linearly independent matrices that span this space.

STEP 2

1. Define the general form of a $2 \times 2$ half-magic matrix.
2. Express the conditions for the matrix to be half-magic.
3. Determine the dimension of the vector space.
4. Find a basis for the vector space.

STEP 3

Let's define a general $2 \times 2$ matrix as:
$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

STEP 4

For the matrix to be half-magic, the sum of the elements in each row must equal the sum of the elements in each column. This gives us the following equations:
1. Row sums: $a + b = c + d$
2. Column sums: $a + c = b + d$

STEP 5

From the conditions $a + b = c + d$ and $a + c = b + d$ , we can express two variables in terms of the other two. Let's solve these equations:
From $a + b = c + d$ , we can express $d$ as: $d = a + b - c$
From $a + c = b + d$ , substitute $d = a + b - c$ : $a + c = b + (a + b - c)$ $a + c = 2b + a - c$ $2c = 2b$ $c = b$
Thus, we have: $d = a + b - b = a$ $c = b$
So the matrix becomes: $A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}$

STEP 6

The general form of the matrix is now: $A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}$
This matrix can be expressed as a linear combination of the following matrices: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
Thus, a basis $B$ for the vector space of $2 \times 2$ half-magic matrices is: $B = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \right\}$
The basis for the vector space of $2 \times 2$ half-magic matrices is:
$B = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \right\}$

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