When does the product \mathbf{AB} exist?

When the number of columns of \mathbf A equals the number of rows of \mathbf B.

How do you find an entry of \mathbf{AB}?

Multiply a row of \mathbf A by a column of \mathbf B: multiply corresponding entries, then add.

\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}e&f\\ g&h\end{pmatrix}?

\begin{pmatrix}ae+bg&af+bh\\ ce+dg&cf+dh\end{pmatrix}

Is matrix multiplication commutative?

No: in general \mathbf{AB}\neq\mathbf{BA}.

State two properties that matrix multiplication does have.

Associative, (\mathbf{AB})\mathbf C=\mathbf A(\mathbf{BC}), and distributive, \mathbf A(\mathbf B+\mathbf C)=\mathbf{AB}+\mathbf{AC}.

What is the 2\times2 identity matrix?

\begin{pmatrix}1&0\\0&1\end{pmatrix}

\mathbf A (and \mathbf{IA}=\mathbf A).

Is \mathbf A^2 found by squaring each entry of \mathbf A?

No: \mathbf A^2=\mathbf{AA}, a row-by-column product.

Matrix arithmeticEdexcel International A Level Further Maths: Flashcards

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What is the order of a matrix with 2 rows and 3 columns?

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What is the order of a matrix with 2 rows and 3 columns?: $2\times3$ (rows first).
When can two matrices be added?: When they have the same order.
How do you add two matrices?: Add corresponding entries.
What does multiplying a matrix by a scalar $k$ do?: Multiplies every entry by $k$ .
When does the product $\mathbf{AB}$ exist?: When the number of columns of $\mathbf A$ equals the number of rows of $\mathbf B$ .
How do you find an entry of $\mathbf{AB}$ ?: Multiply a row of $\mathbf A$ by a column of $\mathbf B$ : multiply corresponding entries, then add.
$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}e&f\\ g&h\end{pmatrix}$ ?: $\begin{pmatrix}ae+bg&af+bh\\ ce+dg&cf+dh\end{pmatrix}$
Is matrix multiplication commutative?: No: in general $\mathbf{AB}\neq\mathbf{BA}$ .
State two properties that matrix multiplication does have.: Associative, $(\mathbf{AB})\mathbf C=\mathbf A(\mathbf{BC})$ , and distributive, $\mathbf A(\mathbf B+\mathbf C)=\mathbf{AB}+\mathbf{AC}$ .
What is the $2\times2$ identity matrix?: $\begin{pmatrix}1&0\\0&1\end{pmatrix}$
What is $\mathbf{AI}$ ?: $\mathbf A$ (and $\mathbf{IA}=\mathbf A$ ).
Order of the product of a $2\times2$ matrix and a $2\times1$ matrix?: $2\times1$
Is $\mathbf A^2$ found by squaring each entry of $\mathbf A$ ?: No: $\mathbf A^2=\mathbf{AA}$ , a row-by-column product.

Exam questions on Matrix arithmetic

$\mathbf{A}=\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}$ and $\mathbf{B}=\begin{pmatrix}1&0\\ -2&5\end{pmatrix}$ .
Find $\mathbf{AB}$ .2 marks
$\mathbf{M}=\begin{pmatrix}1&2\\ 0&-1\end{pmatrix}$ and $\mathbf{N}=\begin{pmatrix}3&1\\ 2&-2\end{pmatrix}$ .
Find $\mathbf{MN}-\mathbf{NM}$ .2 marks
$\mathbf{P}=\begin{pmatrix}a&2\\ 1&-1\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}3&b\\ 0&2\end{pmatrix}$ , where $a$ and $b$ are constants.
Given that $\mathbf{P}+2\mathbf{Q}=\begin{pmatrix}10&8\\ 1&3\end{pmatrix}$ , find the values of $a$ and $b$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).