Adding and subtracting matrices

Adding and subtracting matrices

Two matrices can be added or subtracted only when they have the same order. Then every entry of the first matrix has a partner in the same row and column of the second, and each pair is added or subtracted.

\(\begin{pmatrix}a&\color{#1E6FBA}{\mathbf{b}}\\ c&d\end{pmatrix}+\begin{pmatrix}p&\color{#1E6FBA}{\mathbf{q}}\\ r&s\end{pmatrix}=\begin{pmatrix}a+p&\color{#1E6FBA}{\mathbf{b+q}}\\ c+r&d+s\end{pmatrix}\)

The highlighted entries sit in the same position, row 1 and column 2, so they are added together. Subtraction works in the same way: \(\begin{pmatrix}a&b\\ c&d\end{pmatrix}-\begin{pmatrix}p&q\\ r&s\end{pmatrix}=\begin{pmatrix}a-p&b-q\\ c-r&d-s\end{pmatrix}\).

Why the orders must match

A \(2\times3\) matrix and a \(3\times2\) matrix both have six entries, but the entry in row 1, column 3 of the first matrix has no partner, because the second matrix has no third column. Their sum and difference are not defined. Write "not possible, because the orders are different".

Method

1. Check that the matrices have the same order.
2. Work out any brackets first. Without brackets, work from left to right, as with numbers: \(A-B+C=(A-B)+C\).
3. Add or subtract the entries in the same position. Put brackets around negative entries.
4. Write the answer as one matrix of the same order.

Example

Let \(A=\begin{pmatrix}5&-2\\ 0&3\end{pmatrix}\) and \(B=\begin{pmatrix}1&4\\ -6&3\end{pmatrix}\). Both are \(2\times2\).

\(A-B=\begin{pmatrix}5-1&-2-4\\ 0-(-6)&3-3\end{pmatrix}=\begin{pmatrix}4&-6\\ 6&0\end{pmatrix}\)

Reversing the order changes every sign: \(B-A=\begin{pmatrix}-4&6\\ -6&0\end{pmatrix}\). So \(A-B\) and \(B-A\) are different unless \(A=B\). Addition does not depend on the order: \(A+B=B+A\).

Finding an unknown matrix

If \(X+B=A\), subtract \(B\) from both sides, as with numbers: \(X=A-B\). With the matrices above, \(X=\begin{pmatrix}4&-6\\ 6&0\end{pmatrix}\).

Check: \(X+B=\begin{pmatrix}4+1&-6+4\\ 6+(-6)&0+3\end{pmatrix}=\begin{pmatrix}5&-2\\ 0&3\end{pmatrix}=A\).

Check

Subtracting a negative entry adds: \(0-(-6)=6\), not \(-6\). Adding the zero matrix of the same order leaves a matrix unchanged: \(A+0=A\).