Question

Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M

Answer 1

Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}`

Closure axiom:

Take A = `((x, x),(x, x)) ∈ "M"`

B =  `((y, y),(y, y)) ∈ "M"`

Then AB = `((2xy, 2xy),(2xy, 2xy))`

Since x ≠ 0, y ≠ 0

We see that 2xy ≠ 0 and So AB ∈ M.

This shows that M is closed under matrix multiplication.

Commutative axiom:

AB = `((2xy, 2xy),(2xy, 2xy))`

= BA for all A, B ∈ M

Here, Matrix multiplication is commutative ......(though in general, matrix multiplication is not commutative)

Associative axiom:

Since matrix multiplication is associative, this axiom holds goods for M.