What is a scalar matrix and what is a non scalar matrix?

What is a scalar matrix and what is a non scalar matrix?

If a is equal to e (E is the unit matrix), then a is a scalar matrix

Let a be a matrix over a real number field, and prove that if a ^ t a = 0, then a = 0

Let a = (A1, A2,...) ,an)^T
be
A^TA=a1^2+a2^2+…… +an^2=0
thus
a1=a2=…… =an=0
Then a = 0
Or we can see that a'a is a positive semi definite matrix. If it is equal to 0, there must be a = 0

It is proved that all inverses of n * n matrices over real number fields form a group for matrix multiplication

Let the set of all invertible n * n matrices be m. We know that M is nonempty and matrix multiplication is a binary operation. If M forms a group under matrix multiplication, we will prove it one by one because it satisfies four properties of group
(1) The identity matrix I is invertible and an element in M. for any matrix a ∈ m, IA = AI = a, that is, the identity element exists
(2) For any matrix a ∈ m, there is an inverse matrix A ^ (- 1) such that a * a ^ (- 1) = I, that is, the inverse element exists
(3) Matrix multiplication satisfies the associative law, that is, for any matrix A, B, C ∈ m, it satisfies (a * b) * C = a * (b * c)
(4) For any matrix A, B ∈ m, there is (a * b) * (b ^ (- 1) * a ^ (- 1)) = a * (b * B ^ (- 1)) * a ^ (- 1) = a * I * a ^ (- 1) = I, that is, a * B is invertible, so there is a * B ∈ m, that is, the multiplication of matrix elements is closed
In general, M is a group under matrix multiplication