Matrices and Determinants


   
 
Operations on Matrices
Equality of Matrices
 
Two matrices are said to be equal if they have the same order and their corresponding elements are equal.
 
e.g.,
 
 
then a = 1, b = 2, c = 3, d = 4, e = 5 and f = 6.
 
 
Suggested answer:
 
a = 2, d = -1, e = 3
 
Addition of Matrices
 
If A and B are 2 matrices of the same order, then A + B is the sum of the 2 matrices where each element is got by adding corresponding elements of A and B.
 
 
 
                   
 
Important Results
 
Result 1 (Statement only)
 
Matrix Addition is commutative
 
This means that A + B = B + A, where A, B are 2 matrices of the same order.
 
Verification by numerical problems
 
 
 
 
\ A + B = B + A
 
 
 
 
\ A + B = B + A
 
Result 2 (statement only)
 
Matrix addition is associative
 
i.e., if A, B, C are 3 matrices, then (A + B) + C = A + (B + C)
 
Verification by numerical problems
 
 
then show that (A+B)+C = A+(B+C)
 
Suggested answer:
 
 
 
 
 
From (1) and (2) (A + B) + C = A + (B + C)
 
 
verify the associative law.
 
Suggested answer:
 
 
 
 
 
From (1) and (2), (A + B) + C = A + (B + C)
 
\ The associative law is verified.
 
Subtraction of Matrices
 
For two matrices A and B of the same order, we define A-B = A+(-B).
 
Example:
 
 
 
                           
 
Example:
 
 
Suggested answer:
 
We have 3A - 2B = 3A+(-2)B
 
 
Note: If two matrices A and B are not of same order, A + B and A - B are not defined.
 
Multiplication of a matrix by a scalar
 
Let A=[aij] be an m x n matrix and k be any number called a scalar.
 
Then the matrix obtained by multiplying every element of A by k is
 
called the scalar multiple of A by k and is denoted by kA.
 
Thus, kA = [k aij]m x n
 
Example:
 
 
Multiplication of Matrices
 
Let A be a matrix of order mxn.
 
Let B be a matrix of order nxp.
 
Then the product of the matrices A and B is of order mxp.
 
i.e., when we multiply two matrices the number of columns of the first matrix should be equal to the number of rows of the second matrix.
 
Two matrices can be multiplied by using a 'row-column' multiplication technique illustrated below.
 
 
Multiply the elements of the first row (R1) of A by the corresponding elements of the first column (C1) of B and add the products to get the first element of the first row of AB, namely 2a + 3c + 4e. Next, multiply the elements of the first row (R1) of A by the corresponding elements of the second column (C2) of B and add the products to get the second element of the first row of AB, namely 2b + 3d + 4f.
 
The same steps are used to get the second row of AB, namely R2C1 gives us. 5a + 6c + 7e and R2C2 gives the element 5b + 6d + 7f.
 
 
 
 
Properties of Matrix Multiplication
 
Matrix multiplication is not commutative in general.
 
 
Matrix multiplication is associative i.e., (AB)C = A(BC), whenever both sides are defined.
 
Matrix multiplication is distributive over matrix addition i.e.,
 
(i) A(B + C) = AB + BC
 
(ii) (A + B)C = AB + AC
 
whenever both sides of equality are defined.
 
 
Transpose of a Matrix
 
The transpose of a matrix A is got by interchanging its rows and columns and is denoted by A' or AT.
 
 
If A = [aij]mxn is a matrix of order mxn, the transpose of
 
A = A' = [aji]nxm is a matrix of order nxm.
 
Properties of Transpose
 
(AT)T = A
 
(A + B)T = AT + BT, A and B being of the same order.
 
(KA)T = KAT, k be any scalar (real or complex)
 
(AB)T = BT AT; A and B being conformable for the product AB
 
Symmetric Matrix
 
A square matrix A = [aij] is said to be symmetric if its (i,j)th element is the same as its (j,i)th element.
 
i.e aij = aji " i, j
 
 
 
A square matrix A is said to be symmetric, if A = A'
 
Example:
 
 
 
 
are symmetric matrices.
 
Skew-Symmetric Matrix:
 
A square matrix A = [aij] is said to be skew-symmetric if the (i,j)th element of A is the negative of the (j,i)th element of A
 
i.e., if aij = -aji for all i, j.
 
In particular, for i = j, we have
 
 
 
Thus the diagonal elements of a skew symmetric matrix are all zero.
 
 
 
Properties of Symmetric and Skew Symmetric Matrices
 
1. A square matrix A is said to be skew-symmetric if A' = -A.
 
2. The diagonal elements of a skew-symmetric matrix are all zero.
 
Examples:
 
 
 
skew-symmetric for every square matrix A.
 
 
That is any square matrix is expressible as the sum of a symmetric matrix and a skew-symmetric matrix.
 
5. A matrix which is both symmetric and skew symmetric is a zero matrix.
 
  
     
   
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