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Determinants are one of the most mechanical topics in JEE maths: expand along a row, watch the signs, get a number. It is easy to forget that they were invented to answer a real question: when do several equations have a common solution? The idea appeared first not in Europe but in Japan, in the work of Seki Takakazu. This post covers what is reliably known about him, what he did with determinants, and a worked example of the problem they solve.
Who was Seki Takakazu?
Less is known than you might expect. The MacTutor biography gives his birth as March 1642 and his death as 5 December 1708 in Edo, now Tokyo. Japan's National Diet Library exhibition on Edo-period mathematics is more cautious: it says his year of birth is unknown and confirms that he died in 1708. Both agree on his career. He worked as an accounts official for the Tokugawa lord of the Koshu domain, and when that lord became heir to the shogun, Seki became a shogunate retainer.
According to the National Diet Library, Seki published only one book in his lifetime, *Hatsubi sanpo* (1674), which solved 15 problems left by another mathematician. His other works, including *Kaifukudai no ho*, circulated as handwritten copies among his students. By the late 18th century, Japanese mathematicians called him *Sansei*, the sage of mathematics, and those in his line called themselves the Seki school.
What did Seki do with determinants?
MacTutor's history of matrices and determinants puts it directly: "The idea of a determinant appeared in Japan before it appeared in Europe." In 1683 Seki wrote *Kaifukudai no ho*, translated there as *Method of solving the dissimulated problems*. Without any word for "determinant", he introduced them and gave general methods for calculating them from examples. MacTutor reports that he found determinants of 2 × 2, 3 × 3, 4 × 4 and 5 × 5 arrays and applied them to solving equations, though not to systems of linear equations.
The National Diet Library describes what this was for. Chinese methods could handle equations in one unknown. Seki's notation let several unknowns appear in one expression, which made it easier to eliminate unknowns from simultaneous equations. The exhibition shows a page of *Kaifukudai no ho* that sets out how to calculate a determinant.
Ten years later in Europe, MacTutor notes, Leibniz wrote to l'Hôpital in 1693 with the condition for three linear equations in two unknowns to have a solution, which is exactly the condition that a 3 × 3 determinant is zero. Our post on Arthur Cayley and matrix algebra picks up the story in the 19th century.
Worked example: when do three equations agree?
Find c so that these three equations have a common solution:
x + 2y − 5 = 0
3x − y − 1 = 0
2x + y − c = 0
Write the coefficients as a 3 × 3 array with rows (1, 2, −5), (3, −1, −1) and (2, 1, −c). The equations share a solution only if its determinant is zero. Expand along the first row:
First term: 1 × [(−1)(−c) − (−1)(1)] = 1 × (c + 1) = c + 1.
Second term: −2 × [(3)(−c) − (−1)(2)] = −2 × (−3c + 2) = 6c − 4.
Third term: −5 × [(3)(1) − (−1)(2)] = −5 × 5 = −25.
Total: (c + 1) + (6c − 4) − 25 = 7c − 28. Setting this to zero gives c = 4.
Check without determinants. The first two equations give x = 1, y = 2. Put these into the third: 2 + 2 − c = 0, so c = 4. The determinant has eliminated x and y in one calculation.
What mistakes do students make with determinants?
Dropping the middle sign. Expansion along a row uses signs +, −, +. Step 2 above is where most errors happen.
Mixing up rows and columns. Each row must hold one equation's coefficients in the same order.
Forgetting the constant column. The test for three equations in two unknowns uses the constants as the third column. Leaving them out tests something else.
Our guide to matrices and determinants for JEE has more practice, and the JEE (Main) 2026 syllabus lists determinants of order two and three and the test of consistency. If a sign error keeps creeping in, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student take from Seki's way of working?
MacTutor notes that his 1674 book was remarkable for the careful analysis of each problem, one reason he was so successful as a teacher. He also invented notation when the old one could not say what he needed. Good notation is not decoration: it is what made elimination with many unknowns manageable.
Try it now: find k so that x + y = 3, 2x − y = 0 and kx + y = 4 share a solution, first by substitution and then with a determinant.
