Multiply a matrix by a scalar

Price matrix P in dollars

\(\displaystyle \begin{bmatrix}8&12\\15&20\end{bmatrix}\)

Increase

\(\displaystyle 25\%\)

A print shop raises every price in its price list by 25%. Rows are small and large sizes; columns are plain and printed items. What is the new price of a large printed item, in row 2 column 2?

Move the scalar from positive through zero to negative, and compare the original matrix with its scalar multiple. Watch the signs and keep track of each entry's row and column

What happened to the entries when k went from positive to negative?

Multiplying a matrix by a scalar

To multiply a matrix by a scalar, multiply the scalar by every entry in the matrix. The order and positions of the entries stay unchanged, even when the scalar is negative

1. 1Multiply every entry Multiply the scalar by each entry of the matrix, including entries that are negative or zero \(\displaystyle kA=\begin{bmatrix}ka_{11}&ka_{12}\\ka_{21}&ka_{22}\end{bmatrix}\)
2. 2Keep the order The product has the same number of rows and columns as the starting matrix \(\displaystyle A_{2\times3}\quad\Rightarrow\quad kA_{2\times3}\)
3. 3Apply the sign A negative scalar changes the sign of every non-zero entry, even entries that are already negative \(\displaystyle (-3)(-2)=6,\quad (-3)(4)=-12\)
4. 4Use the multiplier A percentage change is written as a multiplier before it is applied to every entry \(\displaystyle 25\%\text{ increase}:\quad k=1+\frac{25}{100}=1.25\)

Prices P in dollars

\(\displaystyle \begin{bmatrix}45&60\\30&35\end{bmatrix}\)

Winter cut

\(\displaystyle 20\%\)

A pool charges the entry prices in P. Rows are adult and child; columns are weekday and weekend. After a 20% winter cut, what is the winter weekend adult price?

Turn the 20% cut into a multiplier, then apply it to the adult weekend entry of P

Prices P in dollars

\(\displaystyle \begin{bmatrix}45&60\\30&35\end{bmatrix}\)

Next year's prices Q in dollars

\(\displaystyle \begin{bmatrix}54&72\\36&42\end{bmatrix}\)

Next year's prices are Q, where Q = kP for a single scalar k. Find the value of k

Choose a non-zero entry of P and divide the matching entry of Q by it

Scalar

\(\displaystyle -3\)

Matrix A

\(\displaystyle \begin{bmatrix}4&-2\\1&6\end{bmatrix}\)

Applying the scalar to one entry. One of these lines is right. Which one?