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Mathematics | Matrix Introduction

  • Difficulty Level : Easy
  • Last Updated : 24 Jun, 2021

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. 
 

\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}
 

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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 transpose of a matrix: 
 

 

  • (AT)T = A
  • (A+B)T = AT + BT
  • (AB)T = BTAT

 

Singular and Nonsingular Matrix: 
 

 

  1. Singular Matrix: A square matrix is said to be singular matrix if its determinant is zero i.e. |A|=0
  2. Nonsingular Matrix: A square matrix is said to be non-singular matrix if its determinant is non-zero.

 



Properties of Matrix addition and multiplication: 
 

 

  1. A+B = B+A (Commutative)
  2. (A+B)+C = A+ (B+C) (Associative)
  3. AB ? BA (Not Commutative)
  4. (AB) C = A (BC) (Associative)
  5. A (B+C) = AB+AC (Distributive)

 

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 original matrix is equal to its original matrix. i.e. (AT) = A. 
Skew-symmetric: A skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative.i.e. (AT) = -A. 

Diagonal Matrix:A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The term usually refers to square 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 
Involutary Matrix: A matrix is said to be Involutary if A2 = I. 

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). 

Adjoint of a square matrix: The adjoint of a matrix A is the transpose of the cofactor matrix of A

 

\text{If A = }\begin{bmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{bmatrix}\text{then,}
 

 



\text{Adj A = Transpose of }\begin{bmatrix} A_1 & B_1 & C_1\\ A_2 & B_2 & C_2\\ A_3 & B_3 & C_3 \end{bmatrix}\text{=} \begin{bmatrix} A_1 & A_2 & A_3\\ B_1 & B_2 & B_3\\ C_1 & C_2 & C_3 \end{bmatrix}

\text Where, \begin{bmatrix} A_1 & B_1 & C_1\\ A_2 & B_2 & C_2\\ A_3 & B_3 & C_3 \end{bmatrix} \text {is cofactor matrix of A}
 

Properties of Adjoint: 

 

 

  1. A(Adj A) = (Adj A) A = |A| In
  2. Adj(AB) = (Adj B).(Adj A)
  3. |Adj A|= |A|n-1
  4. Adj(kA) = kn-1 Adj(A)
  5. |adj(adj(A))|= |A|^(n-1)^2
  6. adj(adj(A))=|A|^(n-2)     *  A
  7. If A = [L,M,N] then adj(A) = [MN, LN, LM]
  8. adj(I) = I

 

       Where,     “n = number of rows = number of columns”

Inverse of a square matrix: 

 

A^{-1} = \frac{Adj A}{|A|}
 



Here |A| should not be equal to zero, means matrix A should be non-singular. 

Properties of inverse: 

1. (A-1)-1 = A 
2. (AB)-1 = B-1A-1 
3. 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 calculate the inverse of A and by multiplying it B, we can get the values of x, y, and z. 

Trace of a matrix: 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 sum of eigen value of the matrix. For example: 

 

\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} \text{ tr(A) = 1+5+9 = 15}
 

This article is contributed by Nitika Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. 

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

 




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