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IIT JEE Maths Matrices and Determinants

Matrices and Determinants PDF Notes, Important Questions and Synopsis

SYNOPSIS

  1. Determinants
    Every square matrix can be associated to a number called a Determinant.
    For A  → Square matrix; |A| or det A or D → denotes the determinant of A

     
  2. Sub-matrix
    A matrix obtained after deleting some rows or columns is called a sub-matrix.

  3. Minor & Cofactor
    A minor is the determinant of the sub-matrix obained by deleting the ith row and jth column.
    It is denoted by Mij.
    A cofactor is denoted by Cij, and it is given by Cij = (-1)i+j Mij.

  4. Finding Determinant
    A matrix should be a square matrix of order greater than 1, A = [aij]nxn.
    A determinant of a matrix A is defined as the sum of the products of elements of any one row (or column) with corresponding cofact



    |A|= a11C11 + a12C12 + a13C13 (Using first row)ors.

    begin mathsize 12px style OR space vertical line straight A vertical line space equals space straight a subscript 11 open vertical bar table row cell straight a subscript 22 end cell cell straight a subscript 23 end cell row cell straight a subscript 32 end cell cell straight a subscript 33 end cell end table close vertical bar space minus straight a subscript 12 open vertical bar table row cell straight a subscript 21 end cell cell straight a subscript 23 end cell row cell straight a subscript 31 end cell cell straight a subscript 33 end cell end table close vertical bar plus straight a subscript 13 open vertical bar table row cell straight a subscript 21 end cell cell straight a subscript 22 end cell row cell straight a subscript 31 end cell cell straight a subscript 32 end cell end table close vertical bar end style

  5. Properties of a Determinant

    1. The value of a determinant remains the same if the rows and columns are interchanged.begin mathsize 12px style text i.e. D = end text open vertical bar table row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar equals open vertical bar table row cell straight a subscript 1 end cell cell straight b subscript 1 end cell cell straight c subscript 1 end cell row cell straight a subscript 2 end cell cell straight b subscript 2 end cell cell straight c subscript 2 end cell row cell straight a subscript 3 end cell cell straight b subscript 3 end cell cell straight c subscript 3 end cell end table close vertical bar equals straight D ’ rightwards double arrow space straight D space & space straight D ’ space are space transpose space of space each space other. end style
    2. The value of a determinant changes in sign only if any two rows (or columns) of a determinant are interchanged.
      begin mathsize 12px style straight i. straight e. space straight D space equals space open vertical bar table row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar and space straight D ’ space equals open vertical bar table row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar  space straight D ’ space equals space minus space straight D space space or space space straight D space equals space minus space straight D ’ end style
    3. If a determinant has any two rows (or columns) identical, then the value of the determinant is zero.

      begin mathsize 12px style straight D space equals open vertical bar table row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar rightwards double arrow straight D space equals space 0 end style
    4. If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.

      begin mathsize 12px style text i.e. D = end text open vertical bar table row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar and space straight D ’ space open vertical bar table row cell Xa subscript 1 end cell cell Xa subscript 2 end cell cell Xa subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar rightwards double arrow space straight D ’ space equals space XD space end style
    5. If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of two determinants.

      begin mathsize 12px style text D = end text open vertical bar table row cell straight a subscript 1 plus straight x end cell cell straight a subscript 2 plus straight y end cell cell straight a subscript 3 plus straight z end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar equals open vertical bar table row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar plus open vertical bar table row straight x straight y straight z row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar equals space straight D subscript 1 space plus space straight D subscript 2 end style
    6. The value of a determinant does not change by adding to the elements of any row (or columns) the same multiples of the corresponding elements of any other row.

      begin mathsize 12px style straight i. straight e. space space straight D space equals space open vertical bar table row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table close vertical bar and space straight D ’ space equals open vertical bar table row cell straight a subscript 1 plus kb subscript 1 end cell cell straight a subscript 2 plus kb subscript 2 end cell cell straight a subscript 3 plus kb subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell row cell straight c subscript 1 plus mb subscript 1 end cell cell straight c subscript 2 plus mb subscript 2 end cell cell straight c subscript 3 plus mb subscript 3 end cell end table close vertical bar rightwards double arrow space straight D space equals space straight D ’ end style

      Note: While applying this property, at least one row or column must remain unchanged.


  6. Cramer’s Rule (Determinant)
    Consider three simultaneous linear equations:
    a1x + b1y + c1z = d1 ,  a2x + b2y + c2z = d2  and  a3x + b3y + c3z = d3
    The solution of the above system of linear equations is :


 Matrices

A matrix is a rectangular array/arrangement of numbers along rows and columns.

Note: A matrix A = [aij], where aij is an element of the ith row and jth column.
         Row number and column number are the same for diagonal elements.

  1. Classification of Matrices

    • Row Matrix                           A = [3   6   -8]
      Matrix having only                       OR
      one row.                                A = [1   2   6  0] 
    •  Column Matrix
      Matrix having only one column.begin mathsize 12px style table row straight A equals cell open square brackets table attributes columnalign left end attributes row 1 row 2 row 3 end table close square brackets end cell or cell open square brackets table attributes columnalign left end attributes row 2 row 0 row cell negative 1 end cell row 3 end table close square brackets end cell end table end style
    •  Square Matrix
      Matrix having same number of rows and columns.begin mathsize 12px style open square brackets table row 3 6 cell negative 8 end cell row 5 straight i 0 row cell negative 4 end cell 9 1 end table close square brackets subscript 3 cross times 3 end subscript end style
    •   Zero (Null) Matrix
      Matrix having all elements equal to zero.
      Matrix having all elements equal to zero.begin mathsize 12px style open square brackets table row 0 0 0 row 0 0 0 row 0 0 0 end table close square brackets end style
    •    Upper Triangular Matrix
      All entries below the main diagonal are zero.
      All entries below the main diagonal are zero.begin mathsize 12px style open square brackets table row 1 6 cell negative 8 end cell row 0 1 5 row 0 0 1 end table close square brackets end style
    •  Lower Triangular Matrix
      All entries above the main diagonal are zero.begin mathsize 12px style open square brackets table row 3 0 0 row 5 straight i 0 row cell negative 4 end cell 9 1 end table close square brackets end style
    •     Diagonal Matrix
      All entries above and below the diagonal are zero.
      All entries above and below the diagonal are zero.begin mathsize 12px style open square brackets table row 3 0 0 row 0 4 0 row 0 0 9 end table close square brackets end style
    •      Identity (Unit) Matrix
      All diagonal entries are one and the rest are zero.
      All diagonal entries are one and the rest are zero.begin mathsize 12px style open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets end style
  2. Operations on Matrices

    • Addition of Matrices
      Order of the matrices must be same. 
    • Subtraction of Matrices
      Order of the matrices must be same.
    • Equality of matrices
      Matrices having same order with each of their elements equal.
    • Transpose of a Matrix
      Matrix obtained after turning rows into columns and vice versa; denoted by AT.
      Matrix obtained after turning rows into columns and vice versa; denoted by AT.
  3. Multiplication of Matrices

    • Multiplication of a matrix with a scalar
      Each element of the matrix is multiplied by the scalar.

    •    Multiplication of two matrices
      Let A and B be two matrices, then A 
      Let A and B be two matrices, then A ´ B is possible only if ‘No. of columns of 1st matrix = No. of rows of 2nd matrix’.

    Process of Multiplication of two Matrices


  4. Inverse of a Matrix

    Inverse of a matrix A is denoted by A-1 such that (matrix)  (inverse of matrix) = I
    i.e. A × A-1 = I or A-1 × A = I. But A × A-1 ≠ A-1 ´ A.
    Steps of finding inverse of a matrix:

    Step I

    Check whether matrix A is singular or non-singular, i.e.
    |A| = 0
    Þ Singular
    |A|   ≠ 0
    Þ Non-singular

    Step II

    If matrix A is non-singular, then find the value of determinant of A and also find the adjoint matrix A.

    Step III

    Use the formula

     begin mathsize 12px style straight A to the power of negative 1 end exponent equals fraction numerator 1 over denominator vertical line straight A vertical line end fraction adj text end text straight A end style



    Note: begin mathsize 12px style open parentheses straight A to the power of negative 1 end exponent close parentheses to the power of negative 1 end exponent equals straight A comma end styleif A is non-singular.
                If A = diag (a11, a22, …., ann), then begin mathsize 12px style text A end text to the power of negative 1 end exponent equals diag open parentheses 1 over straight a subscript 11 comma 1 over straight a subscript 22 comma. .. comma 1 over straight a subscript nn close parentheses end style
  5. Type of Square matrices

    • Nilpotent Matrix
      If for some least +ve integer p, then Ap =0 is a nilpotent matrix.

      begin mathsize 12px style open square brackets table row 1 2 5 row 2 4 10 row cell negative 1 end cell cell negative 2 end cell cell negative 5 end cell end table close square brackets end style
     Idempotent Matrix
     If A2= A then A is an idempotent matrix.

      begin mathsize 12px style open square brackets table row 2 cell negative 2 end cell cell negative 4 end cell row cell negative 1 end cell 3 4 row 1 cell negative 2 end cell cell negative 3 end cell end table close square brackets end style
       Symmetric Matrix
       If AT = A, then A is a symmetric matrix.
     Skew-symmetric Matrix
     If AT = 
    -A, then A is a skew-symmetric matrix.
     Also, all the diagonal elements are zero.
      Involutory Matrix
      If A2 = I, then A is an involutory matrix.
     Unitary Matrix
     If A’(A’)T = I, where A’ is the complex conjugate of A,   then A is a unitary matrix.
      Orthogonal Matrix
      If A is a square matrix such that ATA = I = ATA or AT = A-1, then A is an orthogonal matrix.

  6. For any square matrix A, A + AT is symmetric and A - AT is skew-symmetric.
  7. Every square matrix can be expressed as a sum of symmetric and skew-symmetric matrices.