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Determinants formula sheet

Every formula in the Determinants chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 28 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Evaluating determinants

Expansion along a row or column
19× asked, last 2025
∣A∣=∑jaijCij,Cij=(−1)i+jMij|A|=\sum_j a_{ij}C_{ij},\qquad C_{ij}=(-1)^{i+j}M_{ij}
Any row or column. Signs: chessboard, ++ at top left, so the row-1 middle term is −a12M12-a_{12}M_{12}. 2×2: ad−bcad-bc.
Cofactors of another row
3× asked, last 2025
∑jaijCkj={∣A∣,k=i0,k≠i\sum_j a_{ij}C_{kj}=\begin{cases}|A|,&k=i\\ 0,&k\ne i\end{cases}
Same for columns. This is A adj⁡A=∣A∣IA\,\operatorname{adj}A=|A|I.
Triangular matrices
3× asked, last 2024
∣A∣=a11a22⋯ann|A|=a_{11}a_{22}\cdots a_{nn}
Upper, lower or diagonal. ∣Ak∣=(a11⋯ann)k|A^k|=(a_{11}\cdots a_{nn})^k.
Determinant as a function of x
14× asked, last 2025
acos⁡θ+bsin⁡θ∈[−a2+b2, a2+b2]a\cos\theta+b\sin\theta\in\left[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}\right]
Simplify, expand, then use this range or calculus. Polynomial Δ(x)\Delta(x): constant term Δ(0)\Delta(0), coefficient sum Δ(1)\Delta(1).
Differentiating
4× asked, last 2025
ddx∣R1R2R3∣=∣R1′R2R3∣+∣R1R2′R3∣+∣R1R2R3′∣\frac d{dx}\begin{vmatrix}R_1\\R_2\\R_3\end{vmatrix}=\begin{vmatrix}R_1^{\prime}\\R_2\\R_3\end{vmatrix}+\begin{vmatrix}R_1\\R_2^{\prime}\\R_3\end{vmatrix}+\begin{vmatrix}R_1\\R_2\\R_3^{\prime}\end{vmatrix}
One row (or column) at a time; every row with xx adds a term. Simplify first, put in xx before expanding.
Sum of determinants
2× asked, last 2024
∑k∣f(k)g(k)h(k)abcpqr∣=∣∑f∑g∑habcpqr∣\sum_k\begin{vmatrix}f(k)&g(k)&h(k)\\a&b&c\\p&q&r\end{vmatrix}=\begin{vmatrix}\sum f&\sum g&\sum h\\a&b&c\\p&q&r\end{vmatrix}
Only when the other rows do not depend on kk.

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22 more formulas in 5 topics

Properties of determinants; Standard determinants; Determinants of kA, adj A and the inverse; Area of a triangle and collinearity; and Systems of linear equations. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Properties of determinants

  • Transpose and swaps5×
  • Scalar factor11×
  • Zero determinant5×
  • Splitting a row4×
  • Row operations17×
  • Product of determinants11×

Standard determinants

  • Equal row sums4×
  • Vandermonde form3×
  • Circulant

Determinants of kA, adj A and the inverse

  • Inverse and adjoint values13×
  • Repeated adjoints11×
  • Nested scalars and adjoints17×
  • Skew-symmetric of odd order2×

Area of a triangle and collinearity

  • Area of a triangle1×
  • Concurrent lines1×

Systems of linear equations

  • Cramer's rule19×
  • Δ = 0: no solution33×
  • All Δᵢ zero
  • Infinitely many solutions44×
  • Homogeneous system14×
  • Two equations in two unknowns4×
  • Matrix method3×

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About 190 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

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