Edexcel IAL - Further Pure Mathematics 1- 5.2 Scalar Multiplication of Matrices- Study notes  - New syllabus

Multiplication of a Matrix by a Scalar

A scalar is a single number. Multiplying a matrix by a scalar means multiplying every entry of the matrix by that number.

Definition

Let \( k \) be a scalar and let

\( A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \)

Then:

\( kA = \begin{pmatrix} ka_{11} & ka_{12} \\ ka_{21} & ka_{22} \end{pmatrix} \)

So every element of the matrix is multiplied by \( k \).

Geometric Interpretation

Scalar multiplication changes the size of a matrix but not its structure.

• If \( k > 1 \), the matrix is enlarged.
• If \( 0 < k < 1 \), the matrix is shrunk.
• If \( k < 0 \), the matrix is also reflected in sign.

Properties

• \( k(A + B) = kA + kB \)
• \( (k + m)A = kA + mA \)
• \( k(mA) = (km)A \)
• \( 1A = A \)