Question

Determine whether w in a subspace of  M_22, given V=M₂₂ and W={invertible 2×2 matrices}

Answer 1

A 2-by-2 matrix is not invertible if its determinant=0.

So:

| a b |

| c d | =ad-bc≠0⇒matrix is invertible.

If we take the sum of two invertible matrices X and Y:

⎛ x₁₁ x₁₂ ⎞ ⎛ y₁₁ y₁₂ ⎞

_⎝ x₂₁ x₂₂ ⎠ ⎝ y₂₁ y₂₂ ⎠=

⎛ x₁₁+y₁₁ x₁₂+y₁₂ ⎞

_⎝ x₂₁+y₂₁ x₂₂+y₂₂ ⎠

This sum must also belong to W for the set to be closed. However, if it can be shown that this is not necessarily an invertible matrix, then the sum doesn't belong to W.

An example of a non-invertible M2×2 is Z=

⎛ 3  4  ⎞

⎝ 9 12 ⎠because its determinant is 3×12-4×9=0

If this matrix can be the sum of 2 invertible matrices then we've found that under matrix addition we can generate a matrix which doesn't belong to W.

Let X and Y be:

⎛ 1 2 ⎞ ⎛ 2 2 ⎞

⎝ 3 4 ⎠ ⎝ 6 8 ⎠

So X+Y=Z and, since X and Y are invertible, because their determinants are non-zero, their sum is Z∉W. So W cannot be a subspace of V (M2×2).

(Incidentally, X^-1=

⎛ -2     1 ⎞

⎝ 3/2 -½ ⎠and

Y^-1=

⎛     2  -½ ⎞

⎝ -3/2   ½ ⎠so they are invertible.)