Matrix Operations

How it works

Add matrices entry by entry when dimensions agree. Matrix multiplication uses row-column dot products and requires the inner dimensions to match. Multiplication order generally matters.

(AB)_{ij}=\sum_k A_{ik}B_{kj}

Worked examples

Example 1

For $\displaystyle A=\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\displaystyle B=\begin{pmatrix}2&0\\1&3\end{pmatrix}$ , find AB.

Row-column products give $\displaystyle AB=\begin{pmatrix}4&6\\1&3\end{pmatrix}$ .

Example 2

Can a $\displaystyle 2\times3$ matrix multiply a $\displaystyle 3\times4$ matrix?

Yes. The inner dimensions agree, and the product has dimensions $\displaystyle 2\times4$ .

Problems

Problem 4.4.1

Compute $\displaystyle \begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}2\\1\end{pmatrix}$ .

Show worked solution

Dot each row with the column: the result is $\displaystyle \begin{pmatrix}4\\10\end{pmatrix}$ .

Problem 4.4.2

Add $\displaystyle \begin{pmatrix}1&-1\\0&2\end{pmatrix}$ and $\displaystyle \begin{pmatrix}3&2\\4&-2\end{pmatrix}$ .

Show worked solution

Entrywise addition gives $\displaystyle \begin{pmatrix}4&1\\4&0\end{pmatrix}$ .

Problem 4.4.3

Why is a $\displaystyle 2\times3$ matrix times a $\displaystyle 2\times2$ matrix undefined?

Show worked solution

The inner dimensions are $\displaystyle 3$ and $\displaystyle 2$ , which do not agree.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.4.1

Problem 4.4.2

Problem 4.4.3

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