Chapter 2 : Matrix

Topics covered in this snack-sized chapter:



Matrix arrow_upward


  • A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.
  • A set of mn numbers arranged in the form of an ordered set of m rows and n columns is called m × n matrix (to be read as m by n matrix).
  • A matrix with m rows and n columns can be written as:

  • Where  represents the element at the intersection of th row and th column.

    • Types of Matrices arrow_upward



    Square Matrix

  • A matrix in which the number of rows is equal to the number of columns is called a Square Matrix.
    • Thus a matrix A with m rows and n columns will be a square matrix if

    m = n.


    Diagonal Elements

  • In a square matrix all those elements  for which  that is, all those elements which occur in the same row and same column namely  are called the diagonal elements.

    • Diagonal Matrix

  • A square matrix A is said to be a diagonal matrix if all its non-diagonal elements be zero.


    • Scalar Matrix

  • A diagonal matrix whose all the diagonal elements are equal, is called a scalar matrix.


    • Unit or Identity Matrix

  • A square matrix A all of whose non-diagonal elements are zero and all the main diagonal elements are unity.

    • Example:

    In general for a unit matrix,

  •  For  and  for

    • Zero Matrix or Null Matrix       

  • Any m × n matrix in which all the elements are zero is called a zero matrix or a null matrix of the type m × n and is denoted by


    • Symmetric Matrix

  • A symmetric matrix is a square matrix that satisfies AT = A where AT represents the transpose such that aij = aji .
  • It also implies A-1 A = I where I is an identity matrix.
  • Let A be the symmetric matrix in which


    • Row Matrix

  • Row Matrix is a matrix with only one row.
  • For example:

    •     []


    Column Matrix

  • A matrix which has only one column is called a Column Matrix.
  • For example:

    •    


    Equal Matrix

  • Two matrices are said to be equal if they have the same order and their corresponding entries are equal.
  • For example:
  •   and  are equal matrix because both the matrix A and B have same order and same corresponding entries.

    • Properties of Matrix Addition arrow_upward


  • Matrix addition is commutative:

    • A + B = B + A

  • Matrix addition is associative:

    •     A + (B + C) = (A + B) + C


    Properties of Matrix Multiplication arrow_upward


  • Multiplication of Matrices is distributive with respect to addition of matrices.

    •     A (B + C) = AB + AC

  • Matrix Multiplication is associative if conformability is assured.
  • A(BC) = (AB)C
  • The multiplication of Matrices is not always commutative.
    • AB is not always equal to BA.
  • Multiplication of a Matrix A by a null matrix conformable with A for multiplication is a null matrix.
    • A0 = 0
  • If AB = 0 then it does not necessarily mean that either A = 0 or B = 0 or both are 0.

    • Example:


    Multiplication of Matrix A by a Unit Matrix

  • Let A be a (m × n) matrix and I be a square unit matrix of order n, so that A and I are conformable for multiplication then

    • Transpose of a Matrix arrow_upward


  • If A be a given matrix of the type m × n then the matrix obtained by changing the rows of A into columns and columns of A into rows is called Transpose of matrix A and is denoted by
  • As there are  rows in  therefore there will be  columns in  and similarly as there are  columns in therefore there will be  rows in

    • Example:

  • Then


    • Properties of Transpose arrow_upward


  • (A')' = A
  • (KA)'= KA’, K being a scalar
  • (A + B)' = A' + B'
  • (AB)' = B'A'
  • (ABC)' = C'B'A'

    • Solution of Linear Equations arrow_upward


  • Consider the set of equations

    • Or

  • The above sets of equations can be conveniently written in matrix form as shown below:

  • That is AX = B

    • Or    

  • That is AX = B
  • In the above, the matrix  is called “coefficient matrix”.
  • If the above equations have a solution, we say that they are consistent and have either a unique solution or infinite solutions.
  • In case they do not have any solution, we can say that the system of equations is inconsistent.

    • Example:

     have a unique solution.

      can be verified by solving them.   

    • Here the coefficient matrix is 

    and

    |A| = -6 - 5 = -11 ≠ 0

    • Matrix  is non-singular and its inverse exists.
    • In this case, we will have a unique solution. The above equation can be written in matrix form as:

    • Where  is a non-singular matrix as: |A| = -11 and

    .

    • Multiplying both sides of (1) by A-1 , we get:

     

    • But A-1 A = I

    • Therefore or

            Or

    Therefore


    Matrix Applications arrow_upward



    Determinant of a Matrix

  • A determinant is a real number associated with every square matrix.
  • The determinant of a square matrix  is denoted by  or.

    • Determinant of a  matrices



    Minors of a Matrix arrow_upward


  • A minor for any element is the determinant that results when the row and column of that element are deleted.
  • For the matrix shown below (Note that R1 is row 1 and C1 is column 1).

    • C1

    C2

    C3

    R1

    1

    4

    3

    R2

    0

    5

    2

     R3

    3

    6

    1


  • Minor for (,, deleted) is:

    • C2

    C3

    R1

    4

    3

    R3

    6

    1



    Matrix of Minors arrow_upward


  • The matrix of minors is the square matrix where each element is the minor for the number in that position.

    • C1

    C2

    C3

    R1

    R2

    R3



    Co-factor of a Matrix arrow_upward


  • Co-factor for any element is either the minor or the opposite of the minor, based on the element's position in the original Determinant.
  • If the row and column of the element add up to an even number.
    • The co-factor is the same as the minor.
  • If the row and column of the element add up to an odd number.
    • The co-factor is the opposite of the minor.

    Co-factor Sign Chart for a 3×3 Matrix arrow_upward



    C1

    C2

    C3

    R1

    +

    -

    +

    R2

    -

    +

    -

    R3

    +

    -

    +



    Adjoint of a Matrix arrow_upward


  • The matrix formed by taking the transpose of the co-factor matrix of a given matrix.
  • The adjoint of matrix is often written as adj A.

    • The Inverse of a 3×3 Matrix arrow_upward


  • It can be found in 6 steps as shown below:
    • Find the matrix of minors.
    • Turn it into a matrix of co-factors (by changing the sign of the odd elements).
    • Find the adjoint by transposing the matrix of co-factors (Columns becomes rows and rows becomes columns).
    • Find the determinant of the matrix.
    • Divide the adjoint of the matrix by the determinant of the matrix.
    • The final result is a matrix inverse.

    Adjoint Method:

     or


    Example:

  • Consider the matrix

  • The co-factor matrix for A is:  
  • So the adjoint is:
  •  
  • Since det A = 22, we get:



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