Let be the set of all real matrices with determinant exactly

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

Let

S

be the set of all real

2\times2

matrices with determinant exactly

1

, together with the usual matrix addition and multiplication by a real scalar. ((a)) Take

\mathsf A = \begin{pmatrix}1&0\\0&1\end{pmatrix}

and

\mathsf B = \begin{pmatrix}0&-1\\1&0\end{pmatrix}

, both of determinant

1

, and compute

\det(\mathsf A+\mathsf B)

. ((b)) Compute

\det(c\mathsf A)

for a general real scalar

c

, and state for which values of

c

(if any, other than

c=1

)

c\mathsf A

stays in

S

. ((c)) Name the requirement or rule (or rules) that fail, and decide whether the zero matrix belongs to

S

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