2.1.3 Scalar Multiple

Scalar Multiples of Matrices

A scalar is simply a real number that multiplies each element of a matrix. When a matrix is multiplied by a scalar, every entry of the matrix is scaled by the same factor.

Scalar Multiplication Rule

Given a scalar $\lambda$ and a matrix

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

The scalar multiplication $\lambda A$ is defined as:

\lambda A = \lambda \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} \lambda a & \lambda b \\ \lambda c & \lambda d \end{pmatrix}

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Example: Scalar Multiplication Let $\lambda = 3$ and

A = \begin{pmatrix} 2 & -1 \\ 4 & 0 \end{pmatrix}

\lambda A = 3 \begin{pmatrix} 2 & -1 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 3 \times 2 & 3 \times -1 \\ 3 \times 4 & 3 \times 0 \end{pmatrix}

= \begin{pmatrix} 6 & -3 \\ 12 & 0 \end{pmatrix}

Matrix Multiplication and Scalar Multiplication Together

Scalar multiplication often appears in combination with matrix multiplication. The key idea is to perform scalar multiplication either before or after matrix multiplication without affecting the result.

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Example: Let

A = \begin{pmatrix} 2 & 1 \\ 0 & -1 \end{pmatrix}, \ B = \begin{pmatrix} -1 & 3 \\ 4 & 0 \end{pmatrix}

And let $\lambda = 2.$

Find $\lambda(AB)$ and $(\lambda A)B$

Step 1: Calculate AB

AB = \begin{pmatrix} 2 & 1 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} -1 & 3 \\ 4 & 0 \end{pmatrix}

= \begin{pmatrix} (2 \times -1) + (1 \times 4) & (2 \times 3) + (1 \times 0) \\ (0 \times -1) + (-1 \times 4) & (0 \times 3) + (-1 \times 0) \end{pmatrix}

= \begin{pmatrix} 2 & 6 \\ -4 & 0 \end{pmatrix}

Step 2: Apply Scalar Multiplication

\lambda(AB) = 2 \begin{pmatrix} 2 & 6 \\ -4 & 0 \end{pmatrix} = \begin{pmatrix} 4 & 12 \\ -8 & 0 \end{pmatrix}

Step 3: Scalar Multiply $A$ First, Then Multiply

\lambda A = 2 \begin{pmatrix} 2 & 1 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 4 & 2 \\ 0 & -2 \end{pmatrix}

Now multiply $(\lambda A)B$

(\lambda A)B = \begin{pmatrix} 4 & 2 \\ 0 & -2 \end{pmatrix} \begin{pmatrix} -1 & 3 \\ 4 & 0 \end{pmatrix}

= \begin{pmatrix} (4 \times -1) + (2 \times 4) & (4 \times 3) + (2 \times 0) \\ (0 \times -1) + (-2 \times 4) & (0 \times 3) + (-2 \times 0) \end{pmatrix}

= \begin{pmatrix} 4 & 12 \\ -8 & 0 \end{pmatrix}

Thus, $\lambda(AB) = (\lambda A)B$

Associativity of Scalar Multiplication

The scalar can also be distributed at any stage:

For matrices $A$ , $B$ , and scalar $\lambda$ ,

\lambda(A \times B) = (\lambda A) \times B = A \times (\lambda B)

This demonstrates that scalar multiplication is associative when combined with matrix multiplication.

Note Summary

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Common Mistakes:

Forgetting to apply the scalar to all elements of the matrix.
Mixing up the order of scalar and matrix multiplication steps. While associative, care is needed to apply the correct sequence in stepwise calculations.
Incorrectly multiplying two matrices first without ensuring compatibility of dimensions.
Misinterpreting scalar matrix equality operations: forgetting the scalar value doesn't change matrix size.

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Key Formulas:

Scalar multiplication:

\lambda \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} \lambda a & \lambda b \\ \lambda c & \lambda d \end{pmatrix}

Associativity of scalar multiplication:

\lambda(A \times B) = (\lambda A) \times B

Scalar distribution:

\lambda(AB) = (A)(\lambda B)

Matrix Multiplication:
$A \times B$ is defined if the inner dimensions match.

Scalar Multiple Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure