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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
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:
$$B = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \right\}$$