Math Infinitum

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Intuition

Two matrices of the same shape add entry by entry, and a number scales every entry at once. It is the arithmetic of vectors, carried over to a rectangular block of numbers.

Two monthly timesheets laid over each other give the hours for both months, cell by cell. Paying double for every hour doubles every cell of the sheet.

Sum and scalar multiple

Two matrices are added only when they have the same shape, and then the entries that share a position are added. Multiplying a matrix by a scalar multiplies every one of its entries. Either way the result has the shape the matrices started with. The difference $A-B$ means $A+(-1)B$ , and the zero matrix $O$ of the right shape changes nothing when it is added.

\begin{aligned}(A+B)_{ij}&=a_{ij}+b_{ij}\\(cA)_{ij}&=c\,a_{ij}\end{aligned}

Rules that follow from the definition

$A+B=B+A$ and $(A+B)+C=A+(B+C)$ , because in each position these are sums of real numbers.

Applications

Practice

Entry by Entry

To add two matrices of the same shape, add the entries in matching positions. To scale a matrix, multiply every entry.

\begin{pmatrix}1&-2\\0&4\end{pmatrix}+\begin{pmatrix}3&5\\-1&2\end{pmatrix}=\begin{pmatrix}4&3\\-1&6\end{pmatrix}

3\begin{pmatrix}1&-2\\0&4\end{pmatrix}=\begin{pmatrix}3&-6\\0&12\end{pmatrix}

Try it

Let $A=\begin{pmatrix}2&-1\\3&0\end{pmatrix}$ and $B=\begin{pmatrix}4&1\\-5&6\end{pmatrix}$ . What is $A+B$ ?

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Only the Entry You Need

One entry of a combination such as $2A-3B$ can be found without computing the rest: in position $(i,j)$ it is $2a_{ij}-3b_{ij}$ .

(2A-3B)_{ij}=2a_{ij}-3b_{ij}

Try it

Let $A=\begin{pmatrix}1&4\\-2&3\end{pmatrix}$ and $B=\begin{pmatrix}0&2\\5&-1\end{pmatrix}$ . What is the entry in row 2, column 1 of $2A-3B$ ?

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Shapes Must Match

Addition pairs up entries in the same position. If one matrix has a position the other lacks, there is nothing to pair it with, and the sum does not exist.

Try it

$A$ is $2\times3$ and $B$ is $3\times2$ . What is $A+B$ ?

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Try it

Let $A=\begin{pmatrix}1&0\\2&-1\end{pmatrix}$ and $B=\begin{pmatrix}0&3\\1&1\end{pmatrix}$ . What is $3A-2B$ ?

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Try it

For any two matrices $A$ and $B$ of the same shape, $A-B=B-A$ .

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Solving for a Matrix

An equation in an unknown matrix $X$ is solved by the same moves as an equation in a number, because each move is made entry by entry.

X+A=B\ \Longrightarrow\ X=B-A

2X=B\ \Longrightarrow\ X=\tfrac12B

Try it

Find $X$ if $2X+\begin{pmatrix}1&3\\0&-2\end{pmatrix}=\begin{pmatrix}5&1\\4&0\end{pmatrix}$ .

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Try it

Find $x$ if $\begin{pmatrix}x&1\\2&0\end{pmatrix}+3\begin{pmatrix}2&0\\1&1\end{pmatrix}=\begin{pmatrix}4&1\\5&3\end{pmatrix}$ .

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Try it

$A$ is a $2\times3$ matrix. What is $A+(-1)A$ ?

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Try it

If $cA=O$ for a scalar $c\neq0$ , then $A=O$ .

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Building a Matrix from Simple Ones

Let $E_{ij}$ be the matrix with $1$ in position $(i,j)$ and $0$ everywhere else. Every matrix is a combination of these, with its own entries as the weights.

\begin{pmatrix}a&b\\c&d\end{pmatrix}=aE_{11}+bE_{12}+cE_{21}+dE_{22}

Try it

Among $2\times2$ matrices, $E_{ij}$ has a $1$ in row $i$ , column $j$ and zeros elsewhere. Which matrix is $3E_{11}-E_{12}+2E_{22}$ ?

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What You Learned

Matrices of the same shape add entry by entry: $(A+B)_{ij}=a_{ij}+b_{ij}$ .

Final checkpoint

Try it

Let $A=\begin{pmatrix}2&0&-1\\1&3&4\end{pmatrix}$ and $B=\begin{pmatrix}1&1&0\\-2&5&2\end{pmatrix}$ . What is the entry in row 2, column 3 of $A-2B$ ?

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Try it

Which rule holds for all matrices $A$ , $B$ of the same shape and all scalars $c$ , $d$ ?

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Try it

The sum $\begin{pmatrix}1&2\\3&4\end{pmatrix}+\begin{pmatrix}1&2&3\end{pmatrix}$ is defined.

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Completion

Lesson complete

Great work! You now know how to:

add and subtract matrices of the same shape;
multiply a matrix by a scalar;
compute one entry of a combination without the rest;
solve simple equations in an unknown matrix.