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Problem 61: Determine whether it is a subspace of the given vector space $\V$
$\V = M_2(\R)$, and $S$ is the subset of all $2\times 2$ matrices with $det(A) = 1$

Problem 61: Determine whether it is a subspace of the given vector space $\V$
$\V = M_2(\R)$, and $S$ is the subset of all $2\times 2$ matrices with $det(A) = 1$.

Solution:
$S = \{A \in M_2(\R) : det(A) = 1\}$
let $k \in R$ be a scalar and let $A \in S$.
Then $det(kA) = k^2\cdot det(A) = k^2\cdot 1 = k^2 \ne 1$, unless $k = ±1$.
Note also that $det(A) = 1$ and $det(B) = 1$ does not imply that $det(A + B) = 1$.
Hence, $S$ is NOT subspace of $M_2(\R)$.

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