MATRICES
A rectangular arrangement of numbers (which may be real or complex
numbers) in rows and columns, is called a matrix.
     A matrix having m rows and n columns is called a matrix of order m×n
     Two matrix A and B are said to be equal matrix if they are of same order and their
corresponding elements are equal
                         1   6   3           𝑎1    𝑎2   𝑎3
     Example: If 𝐴 = [             ]and 𝐵 = [𝑏     𝑏2   𝑏3 ]are equal matrices.
                         5   2   1            1
     Then 𝑎1 = 1, 𝑎2 = 6, 𝑎3 = 3, 𝑏1 = 5, 𝑏2 = 2, 𝑏3 = 1
     (1) Row matrix: A matrix is said to be a row matrix or row vector if it has only one
row. Example: [5 0 3]
   (2) Column matrix: A matrix is said to be a column matrix if it has only one column
           2
Example: [ 3 ]is a column matrix of order 3×1
          −6
      (3) Singleton matrix: If in a matrix there is only one element then it is called
singleton matrix.
      Example: [2], [3], [a], [–3] are singleton matrices.
      (4) Null or zero matrix: If in a matrix all the elements are zero then it is called a
zero matrix and it is generally denoted by O.
                      0 0 0 0 0
      Example: [0], [      ],[          ] , [00] are all zero matrices, but of different orders.
                      0 0 0 0 0
      (5) Square matrix: If number of rows and number of columns in a matrix are equal,
then it is called a square matrix.
                   𝑎11 𝑎12 𝑎13
      Example: [𝑎21 𝑎22 𝑎23 ]is a square matrix of order 3×3
                   𝑎31 𝑎32 𝑎33
       (6) Diagonal matrix: If all elements except the principal diagonal in a square
matrix are zero, it is called a diagonal matrix.
                    2 0 0
      Example : [0 3 0]is a diagonal matrix of order 3×3.
                    0 0 4
      (7) Identity matrix: A square matrix in which elements in the main diagonal are all
'1' and rest are all zero is called an identity matrix or unit matrix.
                                 1 0 0
                      1 0
      Example: [1], [       ] , [0 1 0] are identity matrices of order 1, 2 and 3
                      0 1
                                 0 0 1
respectively.
                                                           Prepared by : Pravesh Kumar, PGT(Maths)
    (8) Scalar matrix: A square matrix whose all non-diagonal elements are zero
and diagonal elements are equal is called a scalar matrix.
                              5 0 0
                     1 0
    Example : [2], [     ] , [0 5 0] are scalar matrices of order 1, 2 and 3
                     0 1
                              0 0 5
respectively.
    (9) Triangular Matrix : A square matrix [𝑎𝑖𝑗 ] is said to be triangular matrix if each
element above or below the principal diagonal is zero. It is of two types
    (i) Upper Triangular matrix :
                3 1 2
    Example : [0 4 3] is an upper triangular matrix of order 3×3.
                0 0 6
    (ii) Lower Triangular matrix :
                1 0 0
    Example : [2 3 0] is a lower triangular matrix of order 3×3.
                4 5 2
     If 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 and 𝐵 = [𝑏𝑖𝑗 ]𝑚×𝑛 are two matrices of the same order then their sum
     A+B is a matrix whose each element is the sum of corresponding elements. i.e.
     𝐴 + 𝐵 = [𝑎𝑖𝑗 + 𝑏𝑖𝑗 ]𝑚×𝑛
                         5 2               1 5                   5+1 2+5          6 7
     Example : If 𝐴 = [1 3] and 𝐵 = [2 2], then 𝐴 + 𝐵 = [1 + 2 3 + 2] = [3 5]
                         4 1               3 3                   4+3 1+3          7 4
     Similarly, their subtraction 𝐴 − 𝐵is defined as 𝐴 − 𝐵 = [𝑎𝑖𝑗 − 𝑏𝑖𝑗 ]𝑚×𝑛
                                    5−1       2−5       4      −3
     i.e. in above example 𝐴 − 𝐵 = [1 − 2     3−2 ] = [−1      1]
                                    4−3       1−3       1      −2
     Let 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 be a matrix and k be a number, then the matrix which is obtained
     by multiplying every element of A by k is called scalar multiplication of A by k and
     it is denoted by kA.
                                                                             2 4
     Thus, if 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 , then 𝑘𝐴 = 𝐴𝑘 = [𝑘𝑎𝑖𝑗 ]𝑚×𝑛 . Example : If 𝐴 = [3 1], then
                                                                             4 6
             10 20
     5𝐴 = [15 5 ]
             20 30
                                                       Prepared by : Pravesh Kumar, PGT(Maths)
     Two matrices A and B are conformable for the product AB if the number of
columns in A (pre-multiplier) is same as the number of rows in B (post multiplier).Thus,
if 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 and 𝐵 = [𝑏𝑖𝑗 ]𝑛×𝑝 are two matrices of order m×n and 𝑛 × 𝑝 respectively,
then their product AB is of order 𝑚 × 𝑝and is defined as (𝐴𝐵)𝑖𝑗 = ∑𝑛𝑟=1 𝑎𝑖𝑟 𝑏𝑟𝑗
                         𝑏1𝑗
                             𝑏
     = [𝑎𝑖1 𝑎𝑖2 . . . 𝑎𝑖𝑛 ] [ 2𝑗
                               ⋮
                                 ]= (ith row of A) (jth column of B) .....(i), where
                             𝑏𝑛𝑗
i=1, 2, ..., m and j=1, 2, ...p
   Now we define the product of a row matrix and a column matrix.
                                                             𝑏1
     Let 𝐴 = [𝑎 𝑎 . . . . 𝑎 ]be a row matrix and 𝐵 = [ 𝑏2 ]be a column matrix.
               1 2        𝑛                           ⋮
                                                𝑏𝑛
    Then 𝐴𝐵 = [𝑎1 𝑏1 + 𝑎2 𝑏2 +. . . . +𝑎𝑛 𝑏𝑛 ]  …(ii). Thus, from (i),
    (𝐴𝐵)𝑖𝑗 =Sum of the product of elements of ith row of A with the corresponding
elements of jth column of B.
    The matrix obtained from a given matrix A by changing its rows into columns or
columns into rows is called transpose of Matrix A and is denoted by 𝐴𝑇 or 𝐴′ .
    From the definition it is obvious that if order of A is m×n, then order of A T is n×m
                                   𝑎1      𝑎2    𝑎3          𝑎1 𝑏1
    Example : Transpose of matrix [𝑏
                                    1      𝑏2    𝑏3 ]2×3 is [𝑎2 𝑏2 ]
                                                             𝑎3 𝑏3 3×2
    Properties of transpose : Let A and B be two matrices then
    (i) (𝐴𝑇 )𝑇 = 𝐴
    (ii) (𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇 , 𝐴and B being of the same order
    (iii) (𝑘𝐴)𝑇 = 𝑘𝐴𝑇 , 𝑘 be any scalar (real or complex)
    (iv) (𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 , 𝐴and B being conformable for the product AB
    (v) 𝐼𝑇 = 𝐼
    (i) Symmetric matrix : A square matrix A is called symmetric matrix if 𝐴𝑇 = 𝐴
               𝑎     ℎ     𝑔
    Example : [ ℎ    𝑏     𝑓]
               𝑔     𝑓     𝑐
      (ii) Skew-symmetric matrix : A square matrix A is called skew- symmetric matrix
                         0    ℎ 𝑔
    𝑇
if 𝐴 = −𝐴. Example : [ −ℎ 0 𝑓 ]
                        −𝑔 −𝑓 0
                                                          Prepared by : Pravesh Kumar, PGT(Maths)
    Note : All principal diagonal elements of a skew- symmetric matrix are always
           zero because.
    (i) If A is a square matrix, then 𝐴 + 𝐴𝑇 is symmetric matrices, while 𝐴 − 𝐴𝑇 is skew-
symmetric matrix.
    (ii) Every square matrix A can uniquelly be expressed as sum of a symmetric and
skew-symmetric matrix i.e.
          1              1
    𝐴 = [ (𝐴 + 𝐴𝑇 )] + [ (𝐴 − 𝐴𝑇 )].
          2              2
    A non-singular square matrix of order n is invertible if there exists a square
matrix B of the same order such that 𝐴𝐵 = 𝐼𝑛 = 𝐵𝐴.
                                                      Prepared by : Pravesh Kumar, PGT(Maths)