Scalar multiplication of a matrix
Scalar multiplication means multiplying every entry of a matrix by the same number, called the scalar. The matrix keeps the same shape; only the entries change.
Method
- Identify the scalar and the matrix.
- Multiply each entry by the scalar, one at a time.
- Simplify each product carefully, including signs and fractions if they appear.
- Rewrite the matrix with the new entries in the same positions.
Example pattern
If the scalar is (k) and the matrix is
[
\begin{bmatrix}
a & b\ c & d
\end{bmatrix},
]
then
[
k\begin{bmatrix}
a & b\ c & d
\end{bmatrix}
\begin{bmatrix}
ka & kb\ kc & kd
\end{bmatrix}.
]
Check your work
- Every original entry should be used exactly once.
- The dimensions of the matrix should not change.
- If the scalar is negative, each entry’s sign should change appropriately.
- If the scalar is a fraction, simplify the final values when possible.
A good final check is to scan row by row and confirm that no entry was skipped or multiplied incorrectly.