PREPWISER.
Practise online

Determinants revision notes

Every idea in the Determinants chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 26 and download the PDF.

Sign in to read all 26Download PDF

Read the first part free. Sign in free with Google to read the rest and download the PDF: 9 A4 pages.9 A4 pages, ready to print.

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

Evaluating determinants, minors and cofactors

Expansion along a row or column
19× asked, last 2025
∣A∣=∑j=1naijCij,Cij=(−1)i+jMij|A|=\sum_{j=1}^{n} a_{ij}C_{ij},\qquad C_{ij}=(-1)^{i+j}M_{ij}
Any row or column gives the same value. MijM_{ij}: delete row ii and column jj. Signs form a chessboard, ++ at top left.
product adad469121827365481total
pairs kk121242121
singular, k2k^2141416414136
++−-++−-++−-++−-++
Chessboard signs
a11a_{11}a12a_{12}a13a_{13}a21a_{21}a22a_{22}a23a_{23}a31a_{31}a32a_{32}a33a_{33}M12M_{12}: delete row 1 and column 2;C12=−M12C_{12}=-M_{12}
Minor and cofactor of a12a_{12}
  • 2×22\times2: ∣A∣=ad−bc|A|=ad-bc. To count singular ones, group ordered pairs (a,d)(a,d) by product: kk pairs with one product give k2k^2 matrices. From {2,3,6,9}\{2,3,6,9\}: product 18 has 4 pairs (16); all products give 36.
  • Non-negative entries: ∣A∣≤|A|\le sum of its ++ terms ≤\le product of the row sums, since that product contains every term. For a 3×33\times3 with all row sums 3, ∣A∣≤27|A|\le27, reached by A=3IA=3I.
  • Not +a12M12+a_{12}M_{12}. The middle term of a row-1 expansion is −a12M12-a_{12}M_{12} because of the chessboard sign. A lost minus flips the answer, and that flipped value is usually an option.
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. In matrix form A (adj A)=∣A∣IA\,(\text{adj}\,A)=|A|I, so ∣A adj A∣=∣A∣n|A\,\text{adj}\,A|=|A|^n for order nn.
  • The cofactor matrix is (adj A)T(\text{adj}\,A)^T, so its determinant is also ∣A∣n−1|A|^{n-1}, and det⁡(A⋅cof A)=∣A∣n\det(A\cdot\text{cof}\,A)=|A|^n even though that product is not ∣A∣I|A|I.
Triangular matrices
3× asked, last 2024
∣A∣=a11 a22⋯ann|A|=a_{11}\,a_{22}\cdots a_{nn}
Upper or lower triangular, or diagonal. So ∣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 with row or column operations, expand, then use calculus or this range. On a restricted domain, read the range there.
  • Polynomial Δ(x)\Delta(x): constant term =Δ(0)=\Delta(0), sum of coefficients =Δ(1)=\Delta(1). For roots in an interval, reduce Δ\Delta to factors first, then count.
Differentiating a determinant
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'\\R_2\\R_3\end{vmatrix}+\begin{vmatrix}R_1\\R_2'\\R_3\end{vmatrix}+\begin{vmatrix}R_1\\R_2\\R_3'\end{vmatrix}
Differentiate one row (or one column) at a time and keep the rest. Put in the value of xx before expanding.
  • Not differentiating only the first row. Every row that contains xx adds its own determinant, because each term of the expansion is a product and needs the product rule.
  • Quick method. Simplify before differentiating. If C1+C2+C3C_1+C_2+C_3 becomes a constant, take it out as a factor; the determinant is constant only if xx then cancels, giving f′=0f'=0.
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}
Valid only when the other rows do not change with kk. Then sum the varying row and simplify once.

Practise Determinants questions as you read. Free, and you can start without signing up.

Practise Determinants

20 more concepts 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.

Sign in to read the restYou come straight back to this spot.Download the PDF

Properties of determinants

  • Transpose and swapping rows5×
  • Taking out a scalar factor11×
  • Zero determinant5×
  • Splitting a row4×
  • Row operations17×
  • Product of determinants11×

Standard determinants

  • Equal row sums4×
  • Vandermonde form3×

Determinants of kA, adj A and the inverse

  • Inverse and adjoint values13×
  • Repeated adjoints11×
  • Nested scalars, adjoints and inverses17×
  • Skew-symmetric of odd order2×

Area of a triangle and collinearity

  • Area of a triangle1×
  • Concurrent lines1×

Systems of linear equations

  • Cramer's rule19×
  • Zero determinant: none or infinitely many33×
  • Infinitely many solutions44×
  • Homogeneous system14×
  • Two equations in two unknowns4×
  • Matrix method3×

Practise Determinants next

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.

Practise Determinants

Also free on PrepWiser

Spotted a mistake? Write to support@prepwiser.in and we will fix it in the next version.

Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

PractiseSign in to read all 26Download PDF
Free with Google, takes a few seconds9 pages, free