Scalar multiple

Definition

A scalar multiple is the result of multiplying a matrix by a scalar (a real number). Each element of the matrix is multiplied by the scalar to produce a new matrix.

5 Must Know Facts For Your Next Test

1. Multiplying a matrix by a scalar results in each element being scaled by that number.
2. The operation does not change the dimensions of the original matrix.
3. Scalar multiplication is distributive, meaning $c(A+B) = cA + cB$ where $c$ is a scalar and $A$, $B$ are matrices.
4. It is also associative with respect to scalar multiplication, i.e., $(ab)C = a(bC)$ where $a$ and $b$ are scalars and $C$ is a matrix.
5. If you multiply any matrix by zero, the result is a zero matrix of the same dimensions.

Review Questions

• What happens to each element of a matrix when it is multiplied by a scalar?
• Is the distributive property applicable in scalar multiplication? Provide an example.
• What is the result when any matrix is multiplied by zero?

Related terms

Matrix Addition:

Operation where two matrices of the same dimensions are added together by adding their corresponding elements.

Identity Matrix:

A square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity for matrices.

Zero Matrix: A matrix in which all elements are zero. It acts as the additive identity for matrices.