That is interesting
Sometimes . . A+
The answer depends on the context. For example, multiplication of numbers is commutative (A*B = B*A) but multiplication of matrices is not.
The Abelian or commutative property of the multiplication of numbers. It is important that both "multiplication" and "numbers" feature in the answer. Because, it is applicable to multiplication but not, for example, for division. It is applicable for the multiplication on numbers but not matrices.
The commutative property works for adding and multiplying e.g. 2+4=4+2 and 3x4=4x3. But it doesn't work for subtraction and division so 5-3≠3-5 and 6÷2≠2÷6 so subtraction and division could be considered as exceptions.
It is so too equal! Multiplication is commutative. Unless A and B are matrices. Matrix multiplication is NOT commutative. Whether or not AxB = BxA depends upon the definition of the binary operator x [multiply] in the domain over which it is defined.
Matrix addition is commutative if the elements in the matrices are themselves commutative.Matrix multiplication is not commutative.
Sometimes . . A+
Subtraction, division, cross multiplication of vectors, multiplication of matrices, etc.
Assuming you mean definition, commutative is a property of an operation such that the order of the operands does not affect the result. Thus for addition, A + B = B + A. Multiplication of numbers is also commutative but multiplication of matrices is not. Subtraction and division are not commutative.
The answer depends on the context. For example, multiplication of numbers is commutative (A*B = B*A) but multiplication of matrices is not.
I assume since you're asking if 2x2 invertible matrices are a "subspace" that you are considering the set of all 2x2 matrices as a vector space (which it certainly is). In order for the set of 2x2 invertible matrices to be a subspace of the set of all 2x2 matrices, it must be closed under addition and scalar multiplication. A 2x2 matrix is invertible if and only if its determinant is nonzero. When multiplied by a scalar (let's call it c), the determinant of a 2x2 matrix will be multiplied by c^2 since the determinant is linear in each row (two rows -> two factors of c). If the determinant was nonzero to begin with c^2 times the determinant will be nonzero, so an invertible matrix multiplied by a scalar will remain invertible. Therefore the set of all 2x2 invertible matrices is closed under scalar multiplication. However, this set is not closed under addition. Consider the matrices {[1 0], [0 1]} and {[-1 0], [0 -1]}. Both are invertible (in this case, they are both their own inverses). However, their sum is {[0 0], [0 0]}, which is not invertible because its determinant is 0. In conclusion, the set of invertible 2x2 matrices is not a subspace of the set of all 2x2 matrices because it is not closed under addition.
The Abelian or commutative property of the multiplication of numbers. It is important that both "multiplication" and "numbers" feature in the answer. Because, it is applicable to multiplication but not, for example, for division. It is applicable for the multiplication on numbers but not matrices.
The multiplication most people are familiar with which you probably learned in school, IS commutative - that's the multiplication of integers, as well as real numbers in general.There are some other operations which mathematicians call "multiplication" which are NOT communitative; for example, the multiplication of matrices, or the cross-product of vectors.
The commutative property works for adding and multiplying e.g. 2+4=4+2 and 3x4=4x3. But it doesn't work for subtraction and division so 5-3≠3-5 and 6÷2≠2÷6 so subtraction and division could be considered as exceptions.
It is so too equal! Multiplication is commutative. Unless A and B are matrices. Matrix multiplication is NOT commutative. Whether or not AxB = BxA depends upon the definition of the binary operator x [multiply] in the domain over which it is defined.
Nothing. Multiplication is commutative and associative.Nothing. Multiplication is commutative and associative.Nothing. Multiplication is commutative and associative.Nothing. Multiplication is commutative and associative.
That is true, matrix multiplication is not commutative.