A system of mn numbers arranged in a rectangular formation along m rows and n columns and bounded by the brackets [ ] is called an m by n matrix; which is written as m × n matrix. A matrix is also represented by a single capital letter A.
Thus, A is matrix of order mn.
It has m rows and n columns. Element of matrix is Each of the mn numbers.
Transpose of a Matrix:
The matrix acquire from any given matrix A, by exchange rows and columns is called the transpose of A and is denoted by AT or A’.
Thus, the transposed matrix of A is A’.
Clearly, the transpose of an m × n matrix is an n × m matrix.
Also, the transpose of the transpose of a matrix matched with itself i.e. (A’)’ = A.
Special Matrices and Properties:
Row and Column Matrix:
- A matrix having a single row is called a row matrix.
- A matrix having a single column is called a column matrix.
- Row and column matrices are sometimes called row vector and column vectors.
- An m × n matrix for which the number of rows is equal to number of columns i.e. m = n, is called square matrix.
- It is also called an n-rowed square matrix.
- The element aij such that i = j, i.e. a11, a22… are called DIAGONAL ELEMENTS and the line along which they line is called Principle Diagonal of matrix.
- Elements other than principal diagonal elements are called off-diagonal elements i.e. aij such that i ≠ j.
A square matrix in which all off-diagonal elements are zero is called a diagonal matrix. The diagonal elements may be zero or may not be zero.
A scalar matrix is a diagonal matrix with all diagonal elements belong equal.
A square matrix each of whose diagonal elements is 1 and each of whose non-diagonal elements are zero is called unit matrix or an identity matrix which is denoted by I.
- Identity matrix is always square.
- Thus, a square matrix A = [aij] is a unit matrix if aij = 1 when i = j and aij = 0 when i ≠ j.
- The m × n matrix whose elements are all zero is called null matrix. Null matrix is denoted by O.
- Null matrix need not be square.
Upper triangular Matrix:
- An upper triangular matrix is a square matrix whose lower off-diagonal elements are zero i.e. aij = 0 whenever i > j.
- It is denoted by U.
- The diagonal and upper off diagonal elements may or may not be zero.
Lower Triangular matrix:
- A lower triangular matrix is a square matrix whose upper off-diagonal triangular elements are zero, i.e., aij = 0 whenever i < j.
- The diagonal and lower off-diagonal elements may or may not be zero. It is denoted by L.
A matrix A is called idempotent if A2 = A.
A matrix A is called involutory if A2 = I.
A matrix A is said to be nilpotent of class m or index m iff Am = 0 and Am – 1 ≠ 0 i.e., m is the smallest index which makes Am = 0
A matrix will be singular matrix if its determinant is equal to zero.
A square matrix A is called periodic if Ak + 1 = A where k is least positive integer and is called the period of A.
Classification of Real Matrices:
Real matrices can be classified into the following three types of the relationship between AT and A.
- A square matrix A = [aij] is said to be symmetric if its (i, j)th elements is same as its (j, i)th element i.e. aij = aij for all i and j.
- In a symmetric matrix: AT = A
Properties of symmetric matrices: For any Square matrix A,
If A and B and symmetric, then:
(a) A + B and A – B are also symmetric
(b) AB, BA maybe symmetric or may not be symmetric.
(c) Ak is symmetric when set of k is any natural number.
(d) AB + BA is symmetric.
(e) AB – BA is skew symmetric.
(f) A2, B2, A2 ± B2 are symmetric.
(g) KA is symmetric where k is scalar quantity.
Skew – Symmetric Matrix:
- A square matrix A = [aij] is said to be skew symmetric if (i, j)th elements of A is the negative of the (j, i)th elements of A if aij = –aij ∀ i, j.
- In a skew symmetric matrix AT = –A.
- A skew symmetric matrix must have all 0’s in the diagonal.
If A and B and symmetric, then:
(a) A ± B are skew symmetric.
(b) AB and BA are not skew symmetric.
(c) A2, B2, A2 ± B2 are symmetric.
(d) A2, A4, A6 are symmetric.
(e) A3, A5, A7 are skew symmetric.
(f) kA is skew symmetric where k is any scalar number.
A square matrix A is said be orthogonal if: AT = A–1 ⇒ AAT = AA–1 = 1. Thus, A will be orthogonal matrix if:
AAT = I = ATA.
Trace of a Matrix:
Let A be a square matrix of order n. The Sum of elements lying in the principal diagonal is called the trace of A denoted by Tr(A).
Thus, if A = [aij]n×n then:
Properties of trace of matrix:
(a). tr (λA) = λ tr(A)
(b). tr (A +B) = tr (A) + tr (B)
(c). tr (AB) = tr (BA)
Inverse of a matrix:
If A be any matrix, then a matrix B if it exists, such that:
AB = BA = I
Then, B is called the Inverse of A which is denoted by A-1 so that AA-1= I.
if A is non-singular matrix.
Properties of Inverse
(a). AA–1 = A–1 A = I
(b). A and B are are inverse of each other iff AB = BA = I
(c). (AB)–1 = B–1 A–1
(d). (ABC)–1 = C–1 B–1 A–1
(e). If A be a n × n non-singular matrix, then (A’)–1 = (A–1)’.
Rank of Matrix:
The rank of a matrix is defined as the order of highest non-zero minor of matrix A. It is denoted by the notation ρ(A). A matrix is said to be of rank r when:
(i) it has at least one non-zero minor of order r, and
(ii) every minor of order higher than r vanishes.
(a). Rank of A and its transpose is the same i.e.
(b). Rank of a null matrix is zero.
(c). Rank of a non-singular square matrix of order r is r.
(d). If a matrix has a non-zero minor of order r, its rank is r and if all minors of a matrix of order r + 1 are zero, its rank is r.
(e). Rank of a matrix is same as the number of linearly independent row vectors vectors in the matrix as well as the number of linearly independent column vectors in the matrix.
(f). For any matrix A, rank (A) min(m,n) i.e. maximum rank of Am×n = min (m, n).
(g). If Rank (AB) Rank A and Rank (AB) Rank B:
so, Rank (AB) ≤ min (Rank A, Rank B)
(h). Rank (AT) = Rank (A)
(i). Rank of a matrix is the number of non-zero rows in its echelon form.
(j). Elementary transformations do not alter the rank of a matrix.
(k). Only null matrix can have a rank of zero. All other matrices have rank of at least one.
(l). Similar matrices have the same rank.
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