SQL Techniques for Performing Operations on Matrices : Page 2

1) Matrix Addition
Matrices A and B can be added, if they are the same size (e.g., m x n). The matrix sum C = A + B also will be a m x n matrix, where for each element c_ij = a_ij + b_ij and 1 ≤ i ≤ m and 1 ≤ j ≤ n.

To add these two matrices:

You would use this solution:

Matrix addition is commutative (e.g., A + B = B + A) and associative (e.g., C + (D + E) = (C + D) + E).

2) Matrix Subtraction
Two matrices (A and B) can be subtracted if they are the same size (e.g., k x l). The matrix difference D = A – B will also be a k x l matrix with the element d_ij = a_ij – b_ij, where 1 ≤ i ≤ k and 1 ≤ j ≤ l.

To subtract matrix B from matrix A:

You would use this solution:

Matrix subtraction is neither commutative nor associative.

3) Matrix Scalar Multiplication
If A is an m x n matrix and the constant k is a number, then the scalar product of k and A is a new matrix B = kA, where b_ij = ka_ij and 1 ≤ i ≤ m and 1 ≤ j ≤ n.

To find B = 5∙A, where:

You would use this solution:

Scalar multiplication is commutative and associative:

k(AB) = (kA)B = A(kB) = (AB)k