Knowledge organisers

Matrices: CBSE Class 12 Maths knowledge organiser

Everything to know about matrices on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Matrix: A rectangular array of numbers with m rows and n columns.
Transpose: Aᵀ is A with its rows and columns swapped.
Symmetric matrix: A square matrix with Aᵀ = A.

Key formulas

Product \(A_{m\times n}B_{n\times p}\) is \(m\times p\); in general \(AB\ne BA\); \(AB=O\) is possible with \(A,B\ne O\)
Transpose	\((A^T)^T=A,\) \((A+B)^T=A^T+B^T,\) \((kA)^T=kA^T,\) \((AB)^T=B^TA^T\)
Symmetric \(A^T=A\); skew-symmetric \(A^T=-A\) (diagonal entries 0)
Any square \(A\)	\(A=\tfrac12(A+A^T)+\tfrac12(A-A^T)\)
Inverse	\(AB=BA=I\Rightarrow B=A^{-1};\) \((AB)^{-1}=B^{-1}A^{-1}\)

Worked example

\(A = \begin{bmatrix}1&3\\0&1\end{bmatrix}\): \(A^2 = \begin{bmatrix}1&6\\0&1\end{bmatrix}\), and in general \(A^n = \begin{bmatrix}1&3n\\0&1\end{bmatrix}\) (each multiplication adds \(3\) to the corner).

Common mistakes

Writing \((AB)^T = A^TB^T\) — the order must reverse.
Expanding \((A + B)^2\) as \(A^2 + 2AB + B^2\) or cancelling: \(AB = AC\) does not give \(B = C\) unless \(A\) is invertible.
Multiplying matrices whose orders do not match, or mixing up rows × columns when stating an order.
Checking only \(AB = I\) when the question asks you to verify an inverse: show \(BA = I\) as well.

You should be able to…

Read and construct matrices, state orders, use equality, add, scale and multiply matrices correctly.
Handle transposes, symmetric/skew-symmetric decomposition, matrix powers and inverses from the definition at board 2- and 3-mark level.
Write complete, step-marked solutions to 5-mark and case-based matrix questions, including context problems.
Solve unfamiliar matrix problems: matrix equations with unknown entries, nilpotent/idempotent reasoning and pattern arguments for powers.

The printable sheet

Matrices knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Matrices knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers