2×2 Matrix Product
Product of two 2×2 matrices.
Formula
Variables
Example
Row times column
How it works
Multiplying two 2×2 matrices combines them by taking each entry of the first matrix row-by-row and each entry of the second matrix column-by-column. Each output entry is a dot product: multiply matching row and column entries, then add them, which is why the top-left entry is ae+bg, the top-right is af+bh, and so on. This works because matrix multiplication represents composition of linear transformations, so the product must capture how one transformation acts after the other.
When to use it
Use this when you need to combine two linear transformations, solve problems involving matrix powers, or compute the result of applying one 2D transformation after another. It also appears in systems of equations, geometry, graphics, and Markov-style state updates.
Worked example, step by step
Let A = [ [2, -1], [3, 4] ] and B = [ [5, 6], [1, -2] ] Multiply corresponding row-column pairs: Top-left: 2·5 + (-1)·1 = 10 - 1 = 9 Top-right: 2·6 + (-1)·(-2) = 12 + 2 = 14 Bottom-left: 3·5 + 4·1 = 15 + 4 = 19 Bottom-right: 3·6 + 4·(-2) = 18 - 8 = 10 So A B = [ [9, 14], [19, 10] ]
Common mistakes to avoid
- Multiplying entries straight across without using row-by-column pairs
- Forgetting to add the two products in each position
- Reversing the order of the matrices, which changes the answer
Did You Know?
Matrix multiplication is not commutative — swapping the order usually gives a different result, unlike ordinary numbers.
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