Math practice · reimagined
Determinants Game
Every grid in this 2×2 determinant game holds four digits from 1 to 9, yet the answer can land anywhere from −80 to 80. Cross the diagonals faster than your rival, and your fighter turns ad − bc into a punch.
Ages 4–99+ · no download · plays in your browser · 23 languages
What is a determinant?
A determinant is a single number worked out from a square matrix, a grid with as many rows as columns. It is written det A or |A|. For a 2×2 matrix, it says how much the grid stretches areas. A determinant of 6 makes every shape six times larger, a negative one also flips it, and 0 squashes it flat.
Example: the determinant of [ 3 5 ; 8 1 ]
- The grid has 3 and 5 on its top row, with 8 and 1 below them.
- Multiply the main diagonal, 3 × 1 = 3, then the other diagonal, 5 × 8 = 40.
- Take the second product away from the first: 3 − 40 = −37.
- The determinant is −37, so you enter it with the minus sign included.
Learn it step by step
A 2×2 matrix is a little grid of four numbers, arranged in two rows: a b on top, c d below. Its determinant is a single number that measures the grid, and there's a tidy cross pattern to find it.
Multiply the main diagonal (top-left × bottom-right = a × d), then subtract the other diagonal (top-right × bottom-left = b × c). For [ 2 3 ; 1 4 ]: that's (2 × 4) − (3 × 1) = 8 − 3 = 5. The answer can come out negative — that's fine, use the − key.
Cross-multiply both diagonals, subtract, and type the determinant.
Tips and common mistakes
- Picture the main diagonal as a slope running downhill from top-left to bottom-right. Downhill product first, uphill product second: that order gives ad − bc every time.
- A common slip is multiplying along the rows, not the diagonals. For [ 2 7 ; 1 5 ], the rows give 14 − 5 = 9, but the diagonals give 10 − 7 = 3.
- Deal four number cards into a 2×2 grid and race each other to its determinant. Then swap the two rows: the answer keeps its size but changes sign.
How to play
Each round shows a 2×2 grid of digits from 1 to 9, like det [ 8 9 ; 8 1 ], and you key in its determinant, −64, minus key included. Level two asks for a row times a column: [2 4] · [2; 6] wants 28. Your rival sees the same problem; be right first and Pea Pod strikes, heavier the quicker you are.
— Quick answers —
Determinant: the short FAQ
- What does a determinant of 0 mean?
- A determinant of 0 means the matrix squashes the plane flat, so every shape loses all its area. It happens when one row is a multiple of the other. In [ 2 3 ; 4 6 ], the bottom row is double the top one, and 2 × 6 − 3 × 4 = 0. Such a matrix has no inverse, the matrix that would undo it.
- How do you multiply matrices?
- You multiply matrices by meeting each row of the first with each column of the second. Pair up their entries, the numbers inside, multiply each pair and add the products: that sum is one entry of the answer. Row [3 4] against column [4; 2] gives 3 × 4 + 4 × 2 = 12 + 8 = 20. The second level asks for exactly one such entry.
- Why do I get the wrong sign on a determinant?
- A wrong sign almost always means the two diagonal products were subtracted in the wrong order. The product from top-left to bottom-right always comes first. In [ 4 9 ; 2 3 ], that gives 12 − 18 = −6, a negative determinant. Start from the other diagonal and you get 6, the right size with the wrong sign.
- At what age do students learn determinants?
- In many school systems, 2×2 determinants arrive in the upper years of secondary school, while others leave them to a first university course in linear algebra. Here, the matrix levels are aimed at 17-year-olds. Students who handle negative numbers and times tables with ease are ready to start.
- What does a class need to play this determinant game, and is it free?
- A class needs only a browser on laptops, tablets or phones, and this linear algebra game is free, with no accounts or installs. For matrix determinant practice, show a calendar month on the board and let each student box a different 2×2 square of dates. Every box gives −7, as 3 × 11 − 4 × 10 shows, and the class can prove why with algebra. On the schools plan, priced per student, a teacher dashboard tracks who is getting quicker.