Scalar multiple of matrix
The multiplication of a vector by a scalar (where the product is a vector), and must be distinguished from inner product of two vectors (where the product is a scalar).
A scalar multiple of a matrix is obtained by multiplying each entry of the matrix by the same scalar quantity. To perform a scalar multiplication on a matrix, simply multiply each entry of the matrix by the scalar value.
For example, let’s consider a matrix A given by:
“`
A = [1 2 3]
[4 5 6]
[7 8 9]
“`
Now, let’s say we want to perform a scalar multiplication on A with a scalar value of 2. To do this, we simply multiply each entry of A by 2, giving us:
“`
2A = [2 4 6]
[8 10 12]
[14 16 18]
“`
Note that the dimensions of the matrix A remain unchanged after scalar multiplication. Also, scalar multiplication is commutative, i.e., kA = Ak, where k is a scalar and A is a matrix.
Scalar multiplication of matrices is used in various mathematical operations, such as matrix addition and subtraction, matrix multiplication, and finding the inverse of a matrix.
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