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Matrices Class 12 Mathematics

Matrices Class 12 Mathematics

1. Basic Concepts

1.1 Definition

A matrix is a collection of numbers arranged into a fixed number of rows and columns. It looks something like this:

$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$

Here, \(a_{11}, a_{12}, a_{21}, a_{22}\) are the elements of the matrix.

1.2 Types of Matrices

  • Square Matrix: A matrix with the same number of rows and columns.
  • Rectangular Matrix: A matrix with a different number of rows and columns.
  • Diagonal Matrix: A square matrix where all elements outside the main diagonal are zero.
  • Scalar Matrix: A diagonal matrix where all the main diagonal elements are the same.
  • Identity Matrix: A diagonal matrix where all the main diagonal elements are 1.
  • Zero Matrix: A matrix where all elements are zero.

2. Important Properties

2.1 Addition and Subtraction

  • Closure Property: The sum or difference of two matrices of the same dimensions is a matrix of the same dimensions.
  • Commutative Property of Addition: \(A + B = B + A\).
  • Associative Property of Addition: \(A + (B + C) = (A + B) + C\).
  • Existence of Additive Identity: Adding a zero matrix to any matrix \(A\) leaves \(A\) unchanged.

2.2 Multiplication

  • Closure Property: The product of two matrices is defined if the number of columns in the first matrix is the same as the number of rows in the second.
  • Associative Property: \(A(BC) = (AB)C\).
  • Distributive Property: \(A(B + C) = AB + AC\) and \((A + B)C = AC + BC\).
  • Existence of Multiplicative Identity: For any square matrix \(A\), \(AI = IA = A\), where \(I\) is the identity matrix of the same order as \(A\).

2.3 Transpose of a Matrix

  • (A^T)^T = A.
  • (A + B)^T = A^T + B^T.
  • (AB)^T = B^T A^T.

2.4 Inverse of a Matrix

  • (A^{-1})^{-1} = A.
  • (AB)^{-1} = B^{-1}A^{-1}.

2.5 Determinant

  • det(A^T) = det(A).
  • det(AB) = det(A)det(B).
  • If A is invertible, then det(A^{-1}) = 1/det(A).
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