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Matrices formula sheet

Every formula in the Matrices chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 25 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.

Types of matrices and operations

Order and types
3× asked, last 2025
A=[aij]m×n;diagonal: aij=0 (i≠j)upper triangular: aij=0 (i>j)\begin{aligned}&A=[a_{ij}]_{m\times n};\quad \text{diagonal: }a_{ij}=0\ (i\ne j)\\&\text{upper triangular: }a_{ij}=0\ (i>j)\end{aligned}
ii = row, jj = column. Scalar matrix =kI=kI. Triangular: AnA^n has diagonal aiina_{ii}^n.
Multiplication
5× asked, last 2022
(AB)ij=∑kaikbkjAm×nBn×p=Cm×p\begin{aligned}&(AB)_{ij}=\sum_k a_{ik}b_{kj}\\&A_{m\times n}B_{n\times p}=C_{m\times p}\end{aligned}
Associative, distributive, AB≠BAAB\ne BA. AB=OAB=O allows A,B≠OA,B\ne O; then square AA has ∣A∣=0|A|=0. XX = column of 1s: XTAXX^TAX = sum of entries.
When matrices commute
4× asked, last 2026
(A+B)2=A2+AB+BA+B2(A+B)2=A2+2AB+B2  ⟺  AB=BA\begin{aligned}&(A+B)^2=A^2+AB+BA+B^2\\&\phantom{(A+B)^2}=A^2+2AB+B^2\iff AB=BA\end{aligned}
Binomial theorem needs AB=BAAB=BA. II, kIkI, A−1A^{-1} and polynomials in AA commute with AA.
Trace
4× asked, last 2026
tr⁡(A±B)=tr⁡A±tr⁡B,tr⁡(kA)=ktr⁡Atr⁡(AB)=tr⁡(BA)\begin{aligned}&\operatorname{tr}(A\pm B)=\operatorname{tr}A\pm\operatorname{tr}B,\quad \operatorname{tr}(kA)=k\operatorname{tr}A\\&\operatorname{tr}(AB)=\operatorname{tr}(BA)\end{aligned}
Square matrices. tr⁡AT=tr⁡A\operatorname{tr}A^T=\operatorname{tr}A, but tr⁡(AB)≠tr⁡A⋅tr⁡B\operatorname{tr}(AB)\ne\operatorname{tr}A\cdot\operatorname{tr}B.

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Practise Matrices

21 more formulas in 5 topics

Transpose, symmetric and skew-symmetric; Powers of a matrix; Adjoint and inverse; Orthogonal and similar matrices; and Counting matrices. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Transpose, symmetric and skew-symmetric

  • Transpose rules4×
  • Symmetric and skew6×
  • Powers and products
  • Counting symmetric and skew2×

Powers of a matrix

  • Idempotent, involutory, nilpotent8×
  • Powers via A = I + N11×
  • Unit triangular 3×3
  • Rotation matrices7×
  • Rank-one matrices2×
  • Periodic powers6×
  • Cayley–Hamilton (2×2)15×
  • Cayley–Hamilton (3×3)

Adjoint and inverse

  • Adjoint3×
  • Adjoint identities8×
  • Inverse5×
  • Inverse from an equation7×
  • Matrix equations7×

Orthogonal and similar matrices

  • Orthogonal matrices9×
  • Similar matrices7×

Counting matrices

  • Trace of AAᵀ5×
  • Other counts11×

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About 110 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 Matrices

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