Matrices · Form 5
Why is matrix multiplication row into column?
Each entry of the product comes from pairing one row of the left matrix with one column of the right, multiplying matching numbers and adding them, which is why the two matrices must line up.
One entry at a time
To find a single number in the answer, you run along a row of the first matrix and down a column of the second at the same time, multiply the numbers that meet, and add the results. That total lands in the position where that row and that column cross.
So a 2 × 2 answer is really four separate row-into-column sums.
The middle numbers must match
For the pairing to work, a row and a column must hold the same count of numbers, so the number of columns in the first matrix must equal the number of rows in the second. Write the orders side by side, say 2 × 3 and 3 × 2: the inner pair (3 and 3) must match for multiplication to be allowed, and the outer pair (2 and 2) gives the order of the answer.
If the inner numbers differ, the product simply does not exist.
Order matters: AB is not BA
The biggest misconception is treating matrices like ordinary numbers, where 3 × 4 equals 4 × 3. With matrices, swapping the order usually changes the answer, and sometimes one order is allowed while the other is not.
It is also not the same as adding: you never just multiply entries in matching positions. Recognise the row-into-column pattern whenever you combine two sets of quantities, such as items against their prices.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How exactly do I calculate one entry of a matrix product?
Take the row from the first matrix and the column from the second matrix that meet at that entry's position. Multiply the matching pairs of numbers together, then add all the products.
Each entry of the answer comes from one full row paired with one full column, never entry-by-entry like addition.
Why can't I multiply two matrices whose orders don't match?
Multiplication pairs each row of the first matrix with each column of the second, matching them position by position. If the number of columns in the first matrix differs from the number of rows in the second, there's nothing to pair with the leftover entries, so the multiplication is undefined.
What's a common mistake when multiplying matrices?
Some students multiply entries position-by-position like addition, rather than pairing a whole row with a whole column and summing. Another common error is assuming AB equals BA, matrix multiplication is usually not commutative, so always multiply in the order the question states.