Matrix Multiplication Calculator
Multiply two 2x2 matrices with row by column steps shown.
What it does
Matrix multiplication turns two tables into one. Each entry of the product dots a row of the first factor against a column of the second, so the answer blends every row with every column. Graphics, economics, and equation solving all lean on this single operation.
The tool above multiplies 2x2 matrices at once. Type both factors to get all four entries with the row by column arithmetic shown. Defaults multiply [[1, 2], [3, 4]] by [[5, 6], [7, 8]], so the page opens solved at 19, 22, 43, 50.
Order is the lesson that sticks. Swapping the factors usually changes the product, unlike ordinary multiplication of numbers. Identity and zero cases frame the behavior: identity returns the other factor untouched (5, 6, 7, 8), while a zero factor wipes everything to zeros.
The example repays a slow read because each entry shows its own dot product: 5 + 14, 6 + 16, 15 + 28, 18 + 32. Once the pattern clicks, larger systems follow the same rule with longer rows and columns.
Use the table for landmark products, then run your matrices. The page fixes both factors at 2x2 with entries inside a wide but finite range. Determinants, inverses, and solvers live on sibling pages; this one exists to make multiplication itself instant and checkable.
How to use
- Enter the four entries of matrix A.
- Enter the four entries of matrix B.
- Read c11 first, then the rest.
- Try swapping A and B to see order bite.
- Test identity and zero cases.
- Carry the product into your system solve.
Formula
- A
- First factor, rows i with entries aij.
- B
- Second factor, columns j with entries bij.
- cij
- Product entry at row i, column j.
Both factors are 2x2 so inner dimensions always agree. Entries accept decimals between -1000000 and 1000000. Results print with up to four decimals, trimming clean integers to whole numbers.
Worked example
Example: 2x2 times 2x2
Multiply A = [[1, 2], [3, 4]] by B = [[5, 6], [7, 8]]. Find all four entries.
- c11 = 1x5 + 2x7 = 5 + 14 = 19.
- c12 = 1x6 + 2x8 = 6 + 16 = 22.
- c21 = 3x5 + 4x7 = 15 + 28 = 43.
- c22 = 3x6 + 4x8 = 18 + 32 = 50.
- Product rows: [19, 22] over [43, 50].
- Swapping order usually changes the answer.
Reference table
| A | B | Product |
|---|---|---|
| I (identity) | [[5, 6], [7, 8]] | [[5, 6], [7, 8]] |
| Zero | [[5, 6], [7, 8]] | [[0, 0], [0, 0]] |
| [[1, 2], [3, 4]] | [[5, 6], [7, 8]] | [[19, 22], [43, 50]] |
| 2I (scaling) | [[5, 6], [7, 8]] | [[10, 12], [14, 16]] |
Limits
- 2x2 factors only.
- Entries inside plus or minus 1000000.
- No determinants or inverses here.
- Plain arithmetic, no step by step solver.
Common mistakes
- Multiplying entry by entry instead.
- Swapping factor order unknowingly.
- Reading c12 where c21 belongs.
- Expecting commutativity from numbers.
- Forgetting rows dot columns.
Try it yourself
Quick checks. Answers sit tucked under each question.
Identity times [[5, 6], [7, 8]]?
c11 5, c12 6, c21 7, c22 8. Identity changes nothing.
Zero matrix times [[5, 6], [7, 8]]?
All four entries are 0.
Diagonal [[2, 0], [0, 3]] times [[5, 6], [7, 8]]? Work it by hand.
c11 = 2x5 + 0x7 = 10. c12 = 2x6 + 0x8 = 12. c21 = 0x5 + 3x7 = 21. c22 = 0x6 + 3x8 = 24.
Questions
How is matrix multiplication done?
Rows of the first times columns of the second: cij = ai1 b1j + ai2 b2j. Each entry dots one row against one column.
Does order matter?
No. A x B usually differs from B x A. The tool multiplies in the order you enter, A first.
What does the identity do?
The identity I = [[1, 0], [0, 1]] leaves any matrix unchanged: I x B = B. The practice run shows 5, 6, 7, 8 back.
What does a zero matrix do?
A zero matrix wipes the product to all zeros regardless of the other factor.
Does it work for other sizes?
Yes, whenever inner dimensions agree. This page fixes both at 2x2 so every entry always exists.
Does it give determinants?
No. Determinants and inverses live on their own pages; this one multiplies only.