Scalar multiples and sums in context

Multiplying a matrix by a scalar

A number that multiplies a matrix is called a scalar. To find \(kA\), multiply every entry of \(A\) by \(k\). The answer has the same order as \(A\). Write the number first, \(kA\), as you write \(3x\).

\(k\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}ka&kb\\ kc&kd\end{pmatrix}\qquad 3\begin{pmatrix}2&-1\\ 0&4\end{pmatrix}=\begin{pmatrix}6&-3\\ 0&12\end{pmatrix}\)

Why: when \(k\) is a positive whole number, \(kA\) is the sum of \(k\) copies of \(A\). For example, \(A+A+A=3A\), and each entry of the sum is three times the entry of \(A\). The same rule is used for every number \(k\): \(-A\) changes the sign of every entry, and \(\tfrac{1}{2}A\) halves every entry.

Combining scalar multiples

Multiply first, then add or subtract, as with numbers. Let \(A=\begin{pmatrix}2&-1\\ 0&4\end{pmatrix}\) and \(B=\begin{pmatrix}1&0\\ -2&3\end{pmatrix}\).

\(2A-3B=\begin{pmatrix}4&-2\\ 0&8\end{pmatrix}-\begin{pmatrix}3&0\\ -6&9\end{pmatrix}=\begin{pmatrix}1&-2\\ 6&-1\end{pmatrix}\)

Sums and scalar multiples of data matrices

A sum or difference of two data matrices is meaningful only when they have the same order and their rows and columns mean the same things, in the same order and with the same unit. A sum or a scalar multiple then keeps those meanings.

A café records the numbers of cups sold on each weekday (Monday to Friday) and on each weekend day (Saturday and Sunday). In both matrices the rows are coffee, tea and the columns are morning, afternoon.

Cups sold in one week: \(5A+2B=\begin{pmatrix}400&250\\ 175&225\end{pmatrix}+\begin{pmatrix}240&180\\ 80&140\end{pmatrix}=\begin{pmatrix}640&430\\ 255&365\end{pmatrix}\).

The 5 counts the weekdays and the 2 counts the weekend days. The difference \(B-A=\begin{pmatrix}40&40\\ 5&25\end{pmatrix}\) shows how many more cups are sold on one weekend day than on one weekday.

Solving for an unknown matrix

1. Move the known matrices to the other side, as with numbers.
2. Divide every entry by the scalar \(k\), which is the same as multiplying by \(\tfrac{1}{k}\).
3. Check by substituting into the original equation.

Example: solve \(2M+\begin{pmatrix}1&3\\ -2&0\end{pmatrix}=\begin{pmatrix}7&-1\\ 4&6\end{pmatrix}\).

\(2M=\begin{pmatrix}7&-1\\ 4&6\end{pmatrix}-\begin{pmatrix}1&3\\ -2&0\end{pmatrix}=\begin{pmatrix}6&-4\\ 6&6\end{pmatrix}\), so \(M=\tfrac{1}{2}\begin{pmatrix}6&-4\\ 6&6\end{pmatrix}=\begin{pmatrix}3&-2\\ 3&3\end{pmatrix}\).

Check: \(2M+\begin{pmatrix}1&3\\ -2&0\end{pmatrix}=\begin{pmatrix}6&-4\\ 6&6\end{pmatrix}+\begin{pmatrix}1&3\\ -2&0\end{pmatrix}=\begin{pmatrix}7&-1\\ 4&6\end{pmatrix}\).

When the unknowns are entries, simplify both sides into single matrices and then equate corresponding entries.

Check

Multiply every entry, including zeros and negative entries. \(A+B\) is one weekday plus one weekend day, not a whole week.

Before adding two data matrices, check that their rows and columns match, then say which period the sum covers (\(5A+2B\) covers Monday to Sunday). To compare two matrices by subtracting, they must cover the same period.