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Matrices revision notes

Every idea in the Matrices chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 22 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 concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Types of matrices and operations

Order and types of matrices
3× asked, last 2025
A=[aij]m×n,diagonal: aij=0 (i≠j),upper triangular: aij=0 (i>j)A=[a_{ij}]_{m\times n},\qquad \text{diagonal: } a_{ij}=0\ (i\ne j),\qquad \text{upper triangular: } a_{ij}=0\ (i>j)
ii = row, jj = column; scalar matrix = kIkI. Triangular: AnA^n has diagonal aiina_{ii}^n.
Matrix multiplication
5× asked, last 2022
(AB)ij=∑kaik bkj,Am×n Bn×p=Cm×p(AB)_{ij}=\sum_{k} a_{ik}\,b_{kj},\qquad A_{m\times n}\,B_{n\times p}=C_{m\times p}
Needs columns of AA = rows of BB. Associative and distributive, but AB≠BAAB\ne BA in general.
123456AA789101112BB5864139154ABAB2×32\times33×23\times22×22\times2row 1 ⋅\cdot column 2: 1⋅8+2⋅10+3⋅12=641\cdot8+2\cdot10+3\cdot12=64
(AB)12(AB)_{12}: row 1 of AA with column 2 of BB; the inner sizes (3 and 3) must match
  • With XX the column of 1s, XTAXX^TAX = sum of all entries of AA. If every row of AA adds to 1, then AX=XAX=X, so AkX=XA^kX=X and every AkA^k also has row sums 1.
  • Not "AB=OAB=O so A=OA=O or B=OB=O". Both can be non-zero. But if AA is square and B≠OB\ne O, then ∣A∣=0|A|=0, since ∣A∣≠0|A|\ne0 would give B=A−1O=OB=A^{-1}O=O; so AX=OAX=O has non-zero solutions.
When matrices commute
4× asked, last 2026
(A+B)2=A2+AB+BA+B2,(A+B)2=A2+2AB+B2  ⟺  AB=BA(A+B)^2=A^2+AB+BA+B^2,\qquad (A+B)^2=A^2+2AB+B^2 \iff AB=BA
The binomial theorem works for AA, BB only if AB=BAAB=BA. II, kIkI, A−1A^{-1} and polynomials in AA commute with AA.
  • Not A2+2AB+B2A^2+2AB+B^2 for (A+B)2(A+B)^2. AB≠BAAB\ne BA in general, so keep AB+BAAB+BA. Expand fully, then compare entries to find the unknowns.
Trace
4× asked, last 2026
tr⁡(A±B)=tr⁡A±tr⁡B,tr⁡(kA)=ktr⁡A,tr⁡(AB)=tr⁡(BA)\operatorname{tr}(A\pm B)=\operatorname{tr}A\pm\operatorname{tr}B,\qquad \operatorname{tr}(kA)=k\operatorname{tr}A,\qquad \operatorname{tr}(AB)=\operatorname{tr}(BA)
Square matrices; tr⁡AT=tr⁡A\operatorname{tr}A^T=\operatorname{tr}A, but tr⁡(AB)\operatorname{tr}(AB) is not the product of traces.

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18 more concepts 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 skew-symmetric6×
  • Counting symmetric and skew matrices2×

Powers of a matrix

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

Adjoint and inverse

  • Adjoint3×
  • Adjoint identities8×
  • Inverse of a matrix5×
  • Inverse from a matrix equation7×
  • Matrix equations AX = B7×

Orthogonal and similar matrices

  • Orthogonal matrices9×
  • Similar matrices7×

Counting matrices

  • Counting by trace of AAᵀ5×
  • Other matrix counts11×

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