Schooling · Class 12 · Mathematics · Matrices · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-I
Matrices: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A matrix is an ordered rectangular…
- Two matrices are equal if and only…
- Addition and subtraction of matrices…
- Scalar multiplication means…
- Matrix multiplication A × B is…
- Matrix multiplication is associative…
- The transpose of a matrix A, written…
- Formulas and facts
- Order of a matrix with m rows and n…
- Element in the iᵗʰ row and jᵗʰ…
- (A + B)′ = A′ + B′
- (kA)′ = kA′ for any scalar k
- (AB)′ = B′A′
- (A′)′ = A
- For a square matrix A
- Watch out for
- Multiplying matrices of incompatible…
- Assuming AB = BA
- Adding matrices of different orders
- Forgetting to reverse order when…
- Confusing a row matrix with a column…
What this chapter is about
Matrices are rectangular arrangements of numbers, symbols or expressions organised in rows and columns. This chapter introduces matrices as mathematical objects that can be added, subtracted, multiplied and transformed according to precise rules. A Class 12 student meets matrices now because they provide a compact way to represent and solve systems of linear equations, perform transformations in geometry, and handle large data sets in economics, physics and computer science.
Key ideas
- A matrix is an ordered rectangular array of numbers arranged in m rows and n columns; its order is written as m × n.
- Two matrices are equal if and only if they have the same order and every corresponding element is equal.
- Addition and subtraction of matrices are defined only for matrices of the same order; you add or subtract corresponding elements.
- Scalar multiplication means multiplying every element of a matrix by the same number (scalar).
- Matrix multiplication A × B is defined only when the number of columns in A equals the number of rows in B; the resulting matrix has order (rows of A) × (columns of B).
- Matrix multiplication is associative and distributive over addition, but it is not commutative: in general, AB ≠ BA.
- The transpose of a matrix A, written Aᵀ or A′, is formed by interchanging its rows and columns.
- A square matrix A is symmetric if A′ = A, and skew-symmetric if A′ = −A; every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.
Formulas and facts to remember
- Order of a matrix with m rows and n columns: m × n; it has m × n elements.
- Element in the iᵗʰ row and jᵗʰ column of matrix A is denoted aᵢⱼ.
- (A + B)′ = A′ + B′ (transpose of a sum equals sum of transposes).
- (kA)′ = kA′ for any scalar k.
- (AB)′ = B′A′ (transpose of a product reverses the order).
- (A′)′ = A (transpose of transpose returns the original matrix).
- For a square matrix A: symmetric part = (1/2)(A + A′); skew-symmetric part = (1/2)(A − A′).
- Identity matrix Iₙ of order n × n has 1 on every diagonal entry and 0 elsewhere; AI = IA = A when multiplication is defined.
- Zero matrix O has every element 0; A + O = A.
Common mistakes
- Multiplying matrices of incompatible orders → check that columns of the first matrix equal rows of the second before multiplying.
- Assuming AB = BA → matrix multiplication is not commutative; always keep the order as given.
- Adding matrices of different orders → addition and subtraction require identical orders; if orders differ, the operation is not defined.
- Forgetting to reverse order when transposing a product → remember (AB)′ = B′A′, not A′B′.
- Confusing a row matrix with a column matrix → a 1 × n matrix (row) and an n × 1 matrix (column) are different objects with different multiplication rules.
Quick revision
- Order m × n means m rows and n columns; total elements = m × n.
- Matrix addition needs same order; matrix multiplication needs (columns of first) = (rows of second).
- AB is generally not equal to BA; always respect the given order.
- Transpose swaps rows and columns; (AB)′ = B′A′.
- Every square matrix = symmetric part + skew-symmetric part.
- Identity matrix I leaves any compatible matrix unchanged under multiplication.