In the world of matrices, we not only deal with operations between matrices but also operations between a matrix and a single number. This single number is commonly referred to as a scalar.
Matrix scalar multiplication is one of the fundamental operations that is important to understand. Imagine you have a cake recipe, and you want to make twice as much. You would naturally multiply each ingredient's measurement by the number 2, right?
A similar concept applies to matrix scalar multiplication.
Matrix scalar multiplication is the operation of multiplying every element in a matrix by a scalar number.
If we have a matrix A and a scalar k, then the result of the scalar multiplication of k by matrix A (written as kA) is a new matrix where each element is the product of the corresponding element of matrix A and the scalar k.
Matrix scalar multiplication has several important properties to be aware of. Let A and B be matrices of the same order, h and k be scalars, and O be the zero matrix.
Distributive over Matrix Addition:
k(A+B)=kA+kB
This means multiplying a scalar by the sum of two matrices is the same as summing the
products of the scalar with each matrix.
Distributive over Scalar Addition:
(h+k)A=hA+kA
This means multiplying the sum of two scalars by a matrix is the same as
summing the products of each scalar with the matrix.
Associative with Scalar Multiplication:
(hk)A=h(kA)=k(hA)
This means multiplying a matrix by the product of two scalars is the same as
multiplying the first scalar by the product of the second scalar and the matrix.
Scalar Multiplication Identity:
1A=A
Multiplying a matrix by the scalar 1 does not change the matrix.
Multiplication by Zero Scalar:
0A=O
Multiplying a matrix by the scalar 0 results in the zero matrix (O), which is a matrix where all elements are 0.
Multiplication of Zero Matrix by a Scalar:
kO=O
Multiplying the zero matrix by any scalar results in the zero matrix.
Multiplication by Scalar −1:
(−1)A=−A
Multiplying a matrix by the scalar −1 results in the negative of the matrix.
These properties help simplify calculations and provide a deeper understanding of matrix algebra.
Since the result of the left side calculation ([627186]) is the same as the result of the right side calculation ([627186]), it is proven that 3(A+B)=3A+3B.
