Click the link below to download the Matrices Assignment Class XII
Download the PDFMatrices 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).
JEE Mains Advance DPP Complex Numbers
Mathematics workbook class 1st
CAT Mathematics sample papers with solution
Class 12 mathematics NCERT Solution
DPP For JEE Mains Advance Trigonometry
HOTS & Important Questions Mathematics class 12
Class 12 mathematics workbook
Chapterwise Test Mathematics Class 12
Mathematics formula book for JEE
Mathematics workbook class 2nd
NCERT Exemplar solution class 12 mathematics
Objective Type Question Bank for Mathematics class 12