YOUR STUDY SPACE · CBSE

Matrices — Class 12 Mathematics

Mathematics Part I · Chapter 3

Download revision card →

CBSE Class 12 community

View all

Mathematics

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Mathematics

~7 pages per kitPage 1 free

Get both kits · ₹79

Physics, Chemistry, Maths + English

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Mathematics
Physics
+ 3 more

~7 pages per kitPage 1 free

Get all 8 kits · ₹199

Physics, Chemistry, Biology + English

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Physics
Chemistry
+ 3 more

~7 pages per kitPage 1 free

Get all 7 kits · ₹199

Accountancy, Business Studies, Economics + English

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Accountancy
Business Studies
+ 3 more

~7 pages per kitPage 1 free

Get all 9 kits · ₹199

Physics

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Physics

~7 pages per kitPage 1 free

Get both kits · ₹79

History

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

History

~7 pages per kitPage 1 free

Get all 3 kits · ₹79

Hindi Elective

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

हिंदी

~7 pages per kitPage 1 free

Get both kits · ₹79

Sociology

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Sociology

~7 pages per kitPage 1 free

Get both kits · ₹79

Economics

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Economics

~7 pages per kitPage 1 free

Get both kits · ₹79

Home Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Home Science

~7 pages per kitPage 1 free

Get both kits · ₹79

Urdu Elective

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

اردو
Urdu Elective

~7 pages per kitPage 1 free

Get both kits · ₹79

Geography

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Geography

~7 pages per kitPage 1 free

Get all 3 kits · ₹79

Accountancy

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Accountancy

~7 pages per kitPage 1 free

Get all 3 kits · ₹79

English Core

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Flamingo
Vistas

~7 pages per kitPage 1 free

Get both kits · ₹79

Political Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Political Science

~7 pages per kitPage 1 free

Get both kits · ₹79

Chemistry

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Chemistry

~7 pages per kitPage 1 free

Get both kits · ₹79

Business Studies

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Business Studies

~7 pages per kitPage 1 free

Get both kits · ₹79

Hindi Core

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

हिंदी

~7 pages per kitPage 1 free

Get both kits · ₹79

संस्कृतम् (केन्द्रिकम्)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

संस्कृत (ऐच्छिक)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

اردو (کور)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Computer Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Fine Art · Painting

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

हिंदुस्तानी संगीत (गायन / वादन)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Biology

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Informatics Practices

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Biotechnology

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

सृजनात्मक लेखन और अनुवाद

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

हिंदुस्तानी संगीत (तबला / पखावज)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

English Elective · Kaleidoscope

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Psychology

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

Space weighted by unit marks
Charts from the book's own data
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹79

Browse all Class 12 kits →

Chapter summary

A matrix is a rectangular array of numbers arranged in rows and columns; its order is . This chapter develops matrix algebra: equality, addition, scalar multiplication and the row-by-column rule for matrix multiplication, where . Matrix multiplication is associative and distributive but generally not commutative, so in general. Special matrices are studied — diagonal, scalar, identity, symmetric () and skew-symmetric (). The transpose operation and its properties, such as , are established. Elementary row and column operations are introduced and used to find the inverse of a matrix. Matrices provide a compact framework for representing and solving systems of linear equations.

Types and order of matricesMatrix operations and multiplicationTranspose, symmetric and skew-symmetric matricesElementary operations and inverse of a matrix

Key ideas & terms

Order of a matrix: The pair giving its number of rows and columns.
Matrix multiplication: ; defined when columns of equal rows of .
Transpose: , obtained by interchanging the rows and columns of .
Symmetric matrix: A square matrix with .
Skew-symmetric matrix: A square matrix with , so its diagonal entries are zero.
Identity matrix: A square matrix with ones on the diagonal and zeros elsewhere; .

Extra questions & answers

When is the product defined?

When the number of columns of equals the number of rows of ; if is and is , then is .

Is matrix multiplication commutative?

No; in general . It is, however, associative and distributive over addition.

Define a symmetric matrix.

A square matrix is symmetric if , i.e. for all .

Define a skew-symmetric matrix.

A square matrix with ; then , forcing all diagonal entries to be zero.

State the transpose of a product.

; the transpose of a product reverses the order of the factors.

How is any square matrix split into symmetric and skew parts?

Write ; the first part is symmetric and the second skew-symmetric.

What is a scalar matrix?

A diagonal matrix whose diagonal entries are all equal; multiplying it by any conformable matrix scales that matrix.

Give the order of if is and is .

The product has order .

What are elementary row operations used for?

They transform a matrix and are used to reduce it to identity form, thereby computing the inverse when it exists.

What is the identity matrix?

A square matrix with s on the main diagonal and s elsewhere; it satisfies .