Question

Show that the set of nonsingluar 2 by 2 matrices is not a vector space. Show also that the set of singular 2 by 2 matrices is not a vector space.

Answer 1

a) 2 nonsingular 2 by 2 matrices are not closed when added together hence it is not a vector space( i.e. their sum = singular and not nonsingular )

b) 2 singular 2 by 2 matrices is not closed under addition, hence they are not a vector space. ( i.e. their sum = nonsingular )

Explanation:

a) Prove that nonsingular 2 by 2 matrices is not a vector space

2 nonsingular matrices are not closed when added together hence it is not a vector space ( i.e. their sum = singular and not nonsingular )

vector A =
`\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]` + vector B =
`\left[\begin{array}{ccc}0&1\\1&0\\\end{array}\right]` =
`\left[\begin{array}{ccc}1&1\\1&1\\\end{array}\right]` ( singular vector )

b) Prove that singular 2 by 2 matrices is not a vector space

2 singular 2 by 2 matrices is not closed under addition, hence they are not a vector space. ( i.e. their sum = nonsingular )

Vector C =
`\left[\begin{array}{ccc}1&0\\0&0\\\end{array}\right]` + vector D =
`\left[\begin{array}{ccc}0&0\\0&1\\\end{array}\right]` =
`\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]` ( nonsingular vector )