1. What Is Scalar Multiplication of a Matrix?
Scalar multiplication means multiplying every element of a matrix by a single real number (called a scalar). It simply scales the entire matrix up or down.
If \( k \) is a real number and \( A = [a_{ij}] \) is a matrix, then the product \( kA \) is obtained by multiplying each entry of \( A \) by \( k \).
\( kA = [k \, a_{ij}] \)
2. How Scalar Multiplication Works
The process is straightforward:
- Take each element of the matrix.
- Multiply it by the scalar.
- Keep the positions of all elements the same.
The size (order) of the matrix does not change after scalar multiplication.
2.1. Example 1
Let \( k = 3 \) and
\( A = \begin{bmatrix} 2 & -1 \\ 4 & 5 \end{bmatrix} \)
Then:
- \( 3 \cdot 2 = 6 \)
- \( 3 \cdot (-1) = -3 \)
- \( 3 \cdot 4 = 12 \)
- \( 3 \cdot 5 = 15 \)
So the result is:
\( 3A = \begin{bmatrix} 6 & -3 \\ 12 & 15 \end{bmatrix} \)
2.2. Example 2
Let \( k = -2 \) and
\( B = \begin{bmatrix} 1 & 3 & -4 \\ 0 & 2 & 5 \end{bmatrix} \)
Multiply each entry by -2:
- \( -2 \cdot 1 = -2 \)
- \( -2 \cdot 3 = -6 \)
- \( -2 \cdot (-4) = 8 \)
- \( -2 \cdot 0 = 0 \)
- \( -2 \cdot 2 = -4 \)
- \( -2 \cdot 5 = -10 \)
Thus:
\( -2B = \begin{bmatrix} -2 & -6 & 8 \\ 0 & -4 & -10 \end{bmatrix} \)
3. Properties of Scalar Multiplication
Scalar multiplication follows these useful properties:
- Distributive over matrix addition: \( k(A + B) = kA + kB \)
- Distributive over scalar addition: \( (k + m)A = kA + mA \)
- Associative with scalars: \( k(mA) = (km)A \)
- Multiplying by 1 leaves the matrix unchanged: \( 1A = A \)
- Multiplying by 0 gives the zero matrix: \( 0A = O \)
These properties make scalar multiplication an essential tool when working with matrix equations and transformations.