Matrix Arithmetic

The Order of a Matrix #

Counted rows first, columns second.

A matrix is a rectangle of numbers, counted rows first and columns second

Read the full lesson →

A matrix is a block of numbers inside brackets, set out in rows and columns.

Rows run across the page. Count them first: this matrix has 2 rows.

Columns run down. Counting them second gives the order 2 × 3.

An entry is named by its row, then its column: row 2, column 3 holds 5.

Adding, Subtracting and Scaling Matrices #

Entry by entry, when the orders match.

Matrices add, subtract and scale entry by entry, so their orders must match

Read the full lesson →

Add entry by entry. The top left of the sum is 3 + 1 = 4.

Every place follows the same rule, so the bottom right is 5 + 2 = 7.

Subtracting works the same way: subtract each entry of B from the matching entry of A.

A 2 × 2 and a 2 × 3 have different orders: the third column has nothing to add to, so this sum cannot be done.

Multiplying by 3 multiplies every entry by 3, and the order never changes.

The Matrix Product #

A row of the left, met with a column of the right.

Each entry of a product comes from multiplying one row of the left matrix by one column of the right

Read the full lesson →

A row of the left matrix holds 3 numbers, and so does a column of the right matrix.

The orders are 2 × 3 and 3 × 2: the inner numbers match, and the outer ones make the product 2 × 2.

Pair row 1 with column 1: multiply each pair and add, 2 × 5 + 3 × 1 + 1 × 3 = 16.

Pair row 1 with column 2 the same way: 2 × 4 + 3 × 0 + 1 × 2 = 10.

Row 2 fills the bottom row the same way, and the product is complete.

Combining Data with a Matrix Product #

A stock table times a price column: takings.

A table of quantities multiplied by a column of prices gives each shop’s total sales

Read the full lesson →

Two shops sold pens and pads. The counts sit in an ordinary table.

Take the labels away and the same counts are a 2 × 2 matrix.

Prices go in a column: $3 for a pen and $5 for a pad.

Shop A sold 4 pens at $3 and 2 pads at $5, so its sales were 4 × 3 + 2 × 5 = $22.

The product is a column of total sales in dollars, one row for each shop.

Why AB Differs from BA #

Swap the order of a product and the answer moves.

Swapping the order of a matrix product usually changes the answer

Read the full lesson →

Multiply each row of A by each column of B to get the product AB.

Now start with B. The same two matrices give a different answer.

So AB and BA are different matrices: matrix multiplication is not commutative.

Orders can differ too: here AB is 2 × 2 while BA is 3 × 3.

The Identity and Zero Matrices #

The matrices that behave like 1 and like 0.

The identity and the zero matrix play the parts that 1 and 0 play for numbers

Read the full lesson →

The identity has ones down the main diagonal and zeros everywhere else.

Multiplying by I changes nothing, exactly as multiplying a number by 1 does.

It works from either side, so with I the order of the product does not matter.

The zero matrix is all zeros, and adding it leaves A exactly as it was.

Multiplying by O gives O, just as multiplying any number by 0 gives 0.

Matrix Powers by Induction #

A pattern in the powers, proved rather than spotted.

A formula for the nth power is proved by multiplying the assumed one by the matrix once more

Read the full lesson →

Work out three powers of this matrix in turn. Only the top-right entry changes.

The pattern suggests a formula for $A^n$Aⁿ, but three cases are not a proof.

The base case: at n = 1 the formula gives a 1 in the corner, which is A itself.

Assume it holds at k. Then the next power is that assumed matrix times A again.

Multiply them out and the corner reads k + 1, which is the formula at n = k + 1.

The base case and the step together prove the formula for every n, exactly as for a sum.