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.
|Text Book||NCERT Based|
|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 = [aij]m x n or A = [aij]
- Let A = [aij] be an m x n matrix. Then A is called
- A row matrix, when m = 1
- Column matrix, when n = 1
- Zero matrix, when aij = 0 for all i, j
- Square matrix, when m = n
- Diagonal matrix, when m = n and aij = 0 for i ≠ j
- A scala, matrix if an only if m = n aij = 0 for all i ≠ j and a11 = a22 = a33 = …………. = ann
- Unit or identity matrix if m = n, aij = 0 for all i ≠ j and aij, 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 AT
- (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 + A1), a symmetric matrix and Q = (A – A1), 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 An is symmetric where ‘n’ is any positive integer.
- Some elementary operations are:
- Ri ↔ Rj ⇒ ith and jth rows are interchanged
- ci ↔ cj ⇒ ith b and jth columns are interchanged
- Ri → k.Ri ⇒ elements of ith row are multiplied by k.
- Ri → Ri + k.Rj ⇒ to the elements of ith row, we add k times, the corresponding elements of the jth row.
- 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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