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A comprehensive review of matrices and determinants, covering essential concepts such as square matrices, transpose properties, special matrix types (orthogonal, idempotent, involutory, nilpotent, singular, and non-singular), and determinant properties. It includes detailed explanations, solved examples from jee main exams, and key theorems like the cayley-hamilton theorem. The material also addresses solving systems of linear equations and explores the relationship between matrices and determinants, making it a valuable resource for high school students preparing for advanced mathematics exams. It also includes differentiation of determinants and some special determinants.
Typology: Summaries
2025/2026
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❖ Square Matrix
INDEX
❖ Algebra of Matrices
❖ Transpose & Its Properties
❖ Some special Types of matrices
❖ Properties of Determinants
❖ Adjoint & Inverse of Matrix
❖ Solving system of Linear equations
❖ Relation between Matrices & Determinants
❖ Cayley Hamilton Theorem
❖ Differentiation of Determinants & Some special Determinants
Matrices
Square Matrix
Square Matrix
Square Matrix
Algebra of Matrices
Algebra of Matrices
Multiplication of a matrix by a scalar
We have, 3A - 2B + 3X = 0 3X = 2B - 3A Solution:Solution:
Algebra of Matrices
Multiplication of two Matrices
A m x n x B n x p = C m x p Pre-multiplier Post multiplier
Algebra of Matrices
- It’s not commutative. i.e. AB ≠ BA (in general)
Properties of Multiplication
Algebra of Matrices
- It’s not commutative. i.e. AB ≠ BA (in general)
- Its Associative i.e. (A × B) × C = A × (B × C)
- It distributes over addition. i.e. A × (B + C) = A × B + A × C or (B + C) × A = B × A + C × A
Properties of Multiplication
Remark
For a matrix A = , if we have to find bigger powers, say A^1000 , one way of doing this is as following: A B C D If M = then which of the following matrices is equal to M^2022?