Mathematics | Matrix Introduction
A matrix represents a collection of numbers arranged in an order of rows and columns. It is necessary to enclose the elements of a matrix in parentheses or brackets.
A matrix with 9 elements is shown below.
This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number. For example, a23 = 6
Order of a Matrix:
The order of a matrix is defined in terms of its number of rows and columns.
Order of a matrix = No. of rows ×No. of columns
Therefore Matrix [M] is a matrix of order 3 × 3.
Transpose of a Matrix :
The transpose [M]T of an m x n matrix [M] is the n x m matrix obtained by interchanging the rows and columns of [M].
if A = [aij] mxn , then AT = [bij] nxm where bij = aji
Properties of the transpose of a matrix:
- (AT)T = A
- (A+B)T = AT + BT
- (AB)T = BTAT
Singular and Nonsingular Matrix:
- Singular Matrix: A square matrix is said to be a singular matrix if its determinant is zero i.e. |A|=0
- Nonsingular Matrix: A square matrix is said to be a non-singular matrix if its determinant is non-zero.
Properties of Matrix addition and multiplication:
- A + B = B + A (Commutative)
- (A + B) + C = A + (B + C) (Associative)
- AB ? BA (Not Commutative)
- (AB) C = A (BC) (Associative)
- A (B+C) = AB + AC (Distributive)
Types of Matrices:
- Square Matrix: A square Matrix has as many rows as it has columns. i.e. no of rows = no of columns.
- Symmetric matrix: A square matrix is said to be symmetric if the transpose of the original matrix is equal to its original matrix. i.e. (AT) = A.
- Skew-symmetric: A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative.i.e. (AT) = -A.
- Diagonal Matrix: A diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The term usually refers to square diagonal matrices.
- Identity Matrix: A square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. Identity matrix is denoted as I.
- Orthogonal Matrix: A matrix is said to be orthogonal if AAT = ATA = I
- Idempotent Matrix: A matrix is said to be idempotent if A2 = A
- Involuntary Matrix: A matrix is said to be Involuntary if A2 = I.
- Zero or Null Matrix: A matrix is said to zero or null matrix if all its elements are zero
- Upper Triangular Matrix: A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix
- Lower Triangular Matrix: A square matrix in which all the elements above the diagonal are zero is known as the lower triangular matrix
Note: Every Square Matrix can uniquely be expressed as the sum of a symmetric matrix and skew-symmetric matrix. A = 1/2 (AT + A) + 1/2 (A – AT).
Determinant of a matrix :
The determinant of a matrix is a number associated with that square matrix. The determinant of a matrix can only be calculated for a square matrix. It is represented by |A|.
Input: 2 X 2 Matrix
Then, the determinant is -> |A| = ad – bc
Adjoint of a square matrix: The adjoint of a matrix A is the transpose of the cofactor matrix of A
Properties of Adjoint:
- A(Adj A) = (Adj A) A = |A| In
- Adj(AB) = (Adj B) . (Adj A)
- |Adj A| = |A|n-1
- Adj(kA) = kn-1 Adj(A)
- |adj(adj(A))| = |A| ^ (n-1) ^ 2
- adj(adj(A)) = |A| ^ (n-2) * A
- If A = [L,M,N] then adj(A) = [MN, LN, LM]
- adj(I) = I
Where, “n = number of rows = number of columns”
The inverse of a square matrix:
Here |A| should not be equal to zero, which means matrix A should be non-singular.
Properties of the inverse:
- (A-1)-1 = A
- (AB)-1 = B-1A-1
- only a non singular square matrix can have an inverse.
Where should we use the inverse matrix?
If you have a set of simultaneous equations:
7x + 2y + z = 21
3y – z = 5
-3x + 4y – 2x = -1
As we know when AX = B, then X = A-1B so we can calculate the inverse of A and by multiplying it by B, we can get the values of x, y, and z.
Trace of a matrix:
The trace of a matrix is denoted as tr(A) which is used only for square matrix and equals the sum of the diagonal elements of the matrix. Remember trace of a matrix is also equal to the sum of the eigenvalue of the matrix. For example:
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