Mastering The 6 Key Properties Of Matrix Addition And Scalar Multiplication: A Must-Know For Linear Algebra And Applied Sciences

6 properties of sum and scalar multiple of matrices

(cd)A=c(dA)c(A+B)=cA+cBc(c+d)A=cA+dA1A = A0 * A = 0c * 0 = 0cA has same dimensions of A

There are six important properties of the sum and scalar multiple of matrices, which are as follows:

1. Closure: The sum and scalar multiple of matrices always results in another matrix of the same size and dimensionality.

2. Associativity: The addition and scalar multiplication of matrices is associative. In other words, if A, B, and C are matrices of the same size, then (A+B)+C=A+(B+C) and (k*m)*A=k*(m*A) for any scalars k and m.

3. Commutativity: Scalar multiplication of matrices is commutative, which means that k*(A+B)=k*A+k*B for any scalar k.

4. Distributivity: Scalar multiplication distributes over matrix addition, which means that both k*(A+B)=k*A+k*B and (k+m)*A=k*A+m*A hold for any scalar k and m and any matrices A and B.

5. Identity element: There exists an identity element in matrix addition, in the sense that there is a matrix O, such that A+O=A for any matrix A. However, there is no identity element in scalar multiplication.

6. Inverse element: Each matrix A has a unique additive inverse -A such that A+(-A)=O, where O is the identity matrix for matrix addition. However, scalar multiplication does not have an inverse element.

In summary, understanding the properties of the sum and scalar multiple of matrices is important for understanding linear algebra and various applications of matrices in science, engineering, and technology.

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