Order and types of matrices, equality, addition, scalar multiplication and multiplication of matrices, transpose, symmetric and skew symmetric matrices, and the idea of an invertible matrix. Our own step-by-step solutions to Exercises 3.1 to 3.4 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. An \(m \times n\) matrix has m rows, n columns and \(mn\) elements; \(a_{ij}\) sits in row i, column j. Two matrices are equal only when they have the same order and every pair of corresponding elements is equal, which turns one matrix equation into several ordinary equations.
What it tests. Add or subtract matrices of the same order element by element; a scalar multiplies every element. The product AB needs (columns of A) = (rows of B), and its (i, j) element is row i of A times column j of B. In general \(AB \ne BA\).
What it tests. \(A'\) swaps rows and columns; \((A \pm B)' = A' \pm B'\), \((kA)' = kA'\), \((AB)' = B'A'\). A is symmetric if \(A' = A\), skew symmetric if \(A' = -A\) (then the diagonal is zero). Every square matrix is \(\tfrac12(A + A') + \tfrac12(A - A')\): symmetric plus skew symmetric.
What it tests. B is the inverse of a square matrix A when \(AB = BA = I\); the inverse, if it exists, is unique, and \((AB)^{-1} = B^{-1}A^{-1}\).
What it tests. Transpose rules (\((AB)' = B'A'\)) settle the symmetric / skew symmetric proofs; equations like \(A'A = I\) or a product equal to O become ordinary equations in the unknowns once the products are multiplied out.
Matrices has 40 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Matrices · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.