Key takeaways
- Each entry of A × B is a row of A times a column of B: multiply pairs and add.
- C₁₁ = ae + bg, C₁₂ = af + bh, C₂₁ = ce + dg, C₂₂ = cf + dh.
- Order matters. A × B is usually different from B × A.
- The calculator's main result is the top-left entry C₁₁. All four entries are in the steps.
In this guide
What is matrix multiplication?
A matrix is a grid of numbers. Multiplying two matrices is not the same as multiplying the matching entries. Instead, each entry of the answer comes from a row of the first matrix and a column of the second.
You multiply them pair by pair and add, just like a dot product.
Why does matrix multiplication matter?
Multiplying matrices is how you combine steps. If one matrix rotates a picture and another stretches it, their product does both at once. Games, animations and robot arms rely on this every frame.
Rotate then stretch is not the same as stretch then rotate. That is exactly why A × B and B × A can be different.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a, b, c, d | Entries of A, row by row | 2, 1, 0, 3 |
| e, f, g, h | Entries of B, row by row | 1, 4, 2, 5 |
| C₁₁ = ae + bg | Row 1 of A · column 1 of B | 4 |
| C₁₂ = af + bh | Row 1 of A · column 2 of B | 13 |
| C₂₁ = ce + dg | Row 2 of A · column 1 of B | 6 |
| C₂₂ = cf + dh | Row 2 of A · column 2 of B | 15 |
The calculator lists all four entries in its steps. The main result shows the top-left entry, C₁₁.
How to multiply 2×2 matrices step by step
- 1Row 1 × column 1. Take (a, b) and (e, g):
a × e + b × g. That is C₁₁. - 2Row 1 × column 2. Take (a, b) and (f, h):
a × f + b × h. That is C₁₂. - 3Row 2 × column 1. Take (c, d) and (e, g):
c × e + d × g. That is C₂₁. - 4Row 2 × column 2. Take (c, d) and (f, h):
c × f + d × h. That is C₂₂. - 5Write the answer. Put the four results in a new 2×2 matrix.
Enter your own two matrices in the calculator below and check all four entries.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Assuming A × B = B × A. Swap the example above and you get [1 4; 2 5] × [2 1; 0 3] = [2 13; 4 17], a different matrix.
- Using a row of B instead of a column. Go down B, not across it.
- Reading only the main result. It is just C₁₁. Look at the steps for the full matrix.
Tips and tricks
- Check with determinants: det(A × B) = det(A) × det(B). Here det(A) = 6, det(B) = −3, and det([4 13; 6 15]) =
60 − 78 = −18. It matches. - Multiply by the identity [1 0; 0 1] and the matrix stays the same, like multiplying a number by 1.
- Check an inverse: A × A⁻¹ should give the identity. Find A⁻¹ with the 2×2 inverse calculator.
Where you'll use it in real life
- Computer graphics and games: combining rotations, stretches and flips.
- Probability: Markov chains, where you multiply a transition matrix by itself to look ahead.
- Algebra: writing systems of equations in matrix form.
- Robotics: chaining the movements of each joint in a robot arm.
Quick summary
To multiply 2×2 matrices, take each row of the first matrix and each column of the second, multiply the pairs and add. Keep the order, because A × B is usually not B × A. Use the calculator above to get all four entries with every step shown.
Worked example
In a graphics exercise you apply two transformations in a row. The first is A = [2 1; 0 3] and the second is B = [1 4; 2 5]. You want the single matrix A × B that does both.
- 01C₁₁ = 2×1 + 1×2 = 4
- 02C₁₂ = 2×4 + 1×5 = 13
- 03C₂₁ = 0×1 + 3×2 = 6
- 04C₂₂ = 0×4 + 3×5 = 15
- 05Product = [4 13; 6 15]
These numbers are pre-filled in the calculator above.
Frequently asked questions
Is matrix multiplication commutative?
No. In general A × B is not equal to B × A, so always keep the matrices in the order given.
What is the identity matrix?
For 2×2 it is [1 0; 0 1]. Multiplying any 2×2 matrix by it leaves the matrix unchanged, like multiplying a number by 1.
Can I multiply any two matrices?
Only if the number of columns in the first equals the number of rows in the second. Two 2×2 matrices always work.
