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

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

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 no matching entries, 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

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

The inner numbers match, and the product of a 2 × 3 and a 3 × 2 is a 2 × 2.

Pair row 1 with column 1: multiply each pair and add, 2 × 5 + 3 × 1 = 13.

Pair row 1 with column 2 the same way: 2 × 4 + 3 × 0 = 8.

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 the takings

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, 5 for a pad.

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

The product is a column of takings, 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

Multiply the rows of A into the columns 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

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

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

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 one step on.

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