Plus Two Maths Notes Chapter 3 Matrices is part of Plus Two Maths Notes. Here we have given Plus Two Maths Notes Chapter 3 Matrices.

Board |
SCERT, Kerala |

Text Book |
NCERT Based |

Class |
Plus Two |

Subject |
Maths Notes |

Chapter |
Chapter 3 |

Chapter Name |
Matrices |

Category |
Kerala Plus Two |

## Kerala Plus Two Maths Notes Chapter 3 Matrices

- A set of mn numbers or functions arranged in the form of a rectangular array of m rows and n columns is called an m X n matrix. In compact form, matrix is represented by A = [a
_{ij}]_{m x n }or A = [a_{ij}] - Let A = [a
_{ij}] be an m x n matrix. Then A is called- A row matrix, when m = 1
- Column matrix, when n = 1
- Zero matrix, when a
_{ij}= 0 for all i, j - Square matrix, when m = n
- Diagonal matrix, when m = n and a
_{ij}= 0 for i ≠ j - A scala, matrix if an only if m = n a
_{ij}= 0 for all i ≠ j and a_{11}= a_{22 }= a_{33 }= …………. = a_{nn } - Unit or identity matrix if m = n, a
_{ij}= 0 for all i ≠ j and a_{ij}, for all i = j

- Let A be an m x n matrix, then the n x m matrix obtained by interchanging the rows and columns of A, is called transpose of A which is denoted by A’ or A
^{T}- (A’)’ =A
- (A + B)’ = A’ + B’
- (kA)’ = kA’
- (AB)’ = B’A’

- A square matrix A is symmetric If A’ = A
- A square matrix A is said to be skew-symmetric if A’ = -A.
- Every square matrix A can be uniquely expressed as A = P + Q where P = (A + A
^{1}), a symmetric matrix and Q = (A – A^{1}), a skew-symmetric matrix. - IfA is a square matrix, then
- (A + A’) is symmetric
- (A – A’) is skew-symmetric
- A.A’ and A’A are symmetric
- Diagonal elements of the skew-symmetric matrix are all zero. ‘
- IfA and B are symmetric matrices of the same order. AB is symmetric if and only if AB = BA. (AB + BA) is symmetric and (AB – BA) is skew-symmetric.
- The determinant of a skew-symmetric main of odd order is zero and of even order is a non-zero perfect square.
- IfA is symmetric, then A
^{n}is symmetric where ‘n’ is any positive integer.

- Some elementary operations are:
- R
_{i}↔ R_{j}⇒ i^{th }and j^{th }rows are interchanged - c
_{i ↔ }c_{j}⇒ i^{th}b and j^{th}columns are interchanged - R
_{i}→ k.R_{i}⇒ elements of i^{th}row are multiplied by k. - R
_{i}→ R_{i}+ k.R_{j}⇒ to the elements of i^{th}row, we add k times, the corresponding elements of the j^{th}row.

- R
- IfA and R are two square matrices such that AB = BA = I, then B is the inverse of A (B = A
^{-1}) and A is the inverse of B. - In erse of a square matrix if it exists is unique.
- Steps to find an inverse of a square matrix A using elementary operations:
- Write A = IA where I – Identity matrix
- Using a sequence of elementary operations, reduce LHS to I. Perform similar operations in t on RHS.
- We obtain I = BA, then B = A
^{-1}

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