Maths-
General
Easy

Question

If A equals open square brackets table row cell fraction numerator 1 over denominator 2 end fraction open parentheses e to the power of i x end exponent plus e to the power of negative i x end exponent close parentheses end cell cell fraction numerator 1 over denominator 2 end fraction open parentheses e to the power of i x end exponent minus e to the power of negative i x end exponent close parentheses end cell row cell fraction numerator 1 over denominator 2 end fraction open parentheses e to the power of i x end exponent minus e to the power of negative i x end exponent close parentheses end cell cell fraction numerator 1 over denominator 2 end fraction open parentheses e to the power of i x end exponent plus e to the power of negative i x end exponent close parentheses end cell end table close square brackets then A to the power of negative 1 end exponent exists

  1. For all real x    
  2. For positive real x only    
  3. For negative real x only    
  4. None of these    

hintHint:

We are given a matrix. We have to find the condition on the variable x so that the inverse of the matrix exist.

The correct answer is: For all real x


    The given matrix is
    A space equals open square brackets table row cell 1 half left parenthesis e to the power of i x end exponent plus e to the power of negative i x end exponent right parenthesis end cell cell 1 half left parenthesis e to the power of i x end exponent minus e to the power of negative i x end exponent right parenthesis end cell row cell 1 half left parenthesis e to the power of i x end exponent minus e to the power of negative i x end exponent right parenthesis end cell cell 1 half left parenthesis e to the power of i x end exponent plus e to the power of i x end exponent right parenthesis end cell end table close square brackets
    The inverse of matrix exists if it's determinant is non zero.
    We will find the determinant of the given matrix.
    D space equals space open vertical bar table row cell 1 half left parenthesis e to the power of i x end exponent plus e to the power of negative i x end exponent right parenthesis end cell cell 1 half left parenthesis e to the power of i x end exponent minus e to the power of negative i x end exponent right parenthesis end cell row cell 1 half left parenthesis e to the power of i x end exponent minus e to the power of negative i x end exponent right parenthesis end cell cell 1 half left parenthesis e to the power of i x end exponent space plus space e to the power of negative i x end exponent right parenthesis end cell end table close vertical bar
space space space equals space 1 fourth left square bracket left parenthesis e to the power of i x end exponent space plus space e to the power of negative i x end exponent right parenthesis left parenthesis e to the power of i x end exponent plus space e to the power of negative i x end exponent right parenthesis right square bracket space minus space 1 fourth left square bracket left parenthesis e to the power of i x end exponent space minus e to the power of negative i x end exponent right parenthesis left parenthesis e to the power of i x end exponent minus e to the power of negative i x end exponent right parenthesis right square bracket
space space space equals space 1 fourth left square bracket left parenthesis e to the power of i x end exponent space plus space e to the power of negative i x end exponent right parenthesis squared space minus left parenthesis e to the power of i x end exponent space minus space e to the power of negative i x end exponent right parenthesis squared right square bracket space space space
space space space equals space 1 fourth left square bracket left parenthesis e to the power of 2 i x end exponent space plus space 2 e to the power of i x end exponent e to the power of negative i x end exponent space plus space e to the power of negative 2 i x end exponent right parenthesis space minus left parenthesis e to the power of 2 i x end exponent space minus 2 e to the power of i x end exponent e to the power of negative i x end exponent space plus space e to the power of negative 2 i x end exponent right parenthesis right square bracket
space space space space equals space 1 fourth left parenthesis e to the power of 2 i x end exponent minus e to the power of 2 i x end exponent plus 2 e to the power of i x minus i x end exponent plus space 2 e to the power of i x minus i x end exponent plus e to the power of negative 2 i x end exponent minus e to the power of negative 2 i x end exponent right parenthesis
space space space space space equals 1 fourth left parenthesis 0 space plus space 2 e to the power of 0 space plus space 2 e to the power of 0 space plus space 0 right parenthesis
space space space space space space equals 1 fourth left parenthesis 2 space plus space 2 right parenthesis space space space space space space... left curly bracket space z e r o space r a i s e d space t o space e space i s space 1 right curly bracket
space space space space space space space equals space 1
    The determinant of the given matrix is non zero. It is also constant. It does not depend on the value of x.
    So, the A-1 exist for all real x.

    Whenever we have to find the inverse of the matrix, we should check the determinant of the matrix. The determinant must be non zero for a inverse to exist.

    Related Questions to study

    General
    Maths-

    If l subscript 1 end subscript superscript 2 end superscript plus m subscript 1 end subscript superscript 2 end superscript plus n subscript 1 end subscript superscript 2 end superscript equals 1 comma etc., and l subscript 1 end subscript l subscript 2 end subscript plus m subscript 1 end subscript m subscript 2 end subscript plus n subscript 1 end subscript n subscript 2 end subscript equals 0 comma etc. and capital delta equals open vertical bar table row cell l subscript 1 end subscript end cell cell m subscript 1 end subscript end cell cell n subscript 1 end subscript end cell row cell l subscript 2 end subscript end cell cell m subscript 2 end subscript end cell cell n subscript 2 end subscript end cell row cell l subscript 3 end subscript end cell cell m subscript 3 end subscript end cell cell n subscript 3 end subscript end cell end table close vertical bar then

    If l subscript 1 end subscript superscript 2 end superscript plus m subscript 1 end subscript superscript 2 end superscript plus n subscript 1 end subscript superscript 2 end superscript equals 1 comma etc., and l subscript 1 end subscript l subscript 2 end subscript plus m subscript 1 end subscript m subscript 2 end subscript plus n subscript 1 end subscript n subscript 2 end subscript equals 0 comma etc. and capital delta equals open vertical bar table row cell l subscript 1 end subscript end cell cell m subscript 1 end subscript end cell cell n subscript 1 end subscript end cell row cell l subscript 2 end subscript end cell cell m subscript 2 end subscript end cell cell n subscript 2 end subscript end cell row cell l subscript 3 end subscript end cell cell m subscript 3 end subscript end cell cell n subscript 3 end subscript end cell end table close vertical bar then

    Maths-General
    General
    Maths-

    If A equals open square brackets table row alpha 0 row 1 1 end table close square brackets and B equals open square brackets table row 1 0 row 3 1 end table close square brackets comma then value of alpha for which A to the power of 2 end exponent equals B is

    For such questions, we should know how to multiply to matrices. When there is equal sign between two matrices, the elements of both the matrix should be equal.

    If A equals open square brackets table row alpha 0 row 1 1 end table close square brackets and B equals open square brackets table row 1 0 row 3 1 end table close square brackets comma then value of alpha for which A to the power of 2 end exponent equals B is

    Maths-General

    For such questions, we should know how to multiply to matrices. When there is equal sign between two matrices, the elements of both the matrix should be equal.

    General
    Maths-

    For the primitive integral equation y d x plus y to the power of 2 end exponent d y equals x d y comma x element of R comma y greater than 0 comma y equals y open parentheses x close parentheses comma y open parentheses 1 close parentheses equals 1 comma then y open parentheses negative 3 close parentheses is

    For such questions, we should know different method of differentiation and integration.

    For the primitive integral equation y d x plus y to the power of 2 end exponent d y equals x d y comma x element of R comma y greater than 0 comma y equals y open parentheses x close parentheses comma y open parentheses 1 close parentheses equals 1 comma then y open parentheses negative 3 close parentheses is

    Maths-General

    For such questions, we should know different method of differentiation and integration.

    parallel
    General
    Maths-

    The differential equation of all circles which pass through the origin and whose centre lies on y-axis is

    For such questions, we should know the equation of cricle with its centre at a point other than origin.

    The differential equation of all circles which pass through the origin and whose centre lies on y-axis is

    Maths-General

    For such questions, we should know the equation of cricle with its centre at a point other than origin.

    General
    Maths-

    The differential equation of all parabolas whose axis are parallel to y-axis is

    For such questions, we should know how to write the equation of a parabola. The axis of parabola can be parallel to x or y axis. So, we have to write the equation accordingly.

    The differential equation of all parabolas whose axis are parallel to y-axis is

    Maths-General

    For such questions, we should know how to write the equation of a parabola. The axis of parabola can be parallel to x or y axis. So, we have to write the equation accordingly.

    General
    Maths-

    Solution of the differential equation open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 2 end exponent minus fraction numerator d y over denominator d x end fraction open parentheses e to the power of x end exponent plus e to the power of negative x end exponent close parentheses plus 1 equals 0 is given by

    When we get multipliction of two brackets equal to zero, any of the bracket can be zero or both of the brackets can be zero. It is decided based on the conditions in the equation.

    Solution of the differential equation open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 2 end exponent minus fraction numerator d y over denominator d x end fraction open parentheses e to the power of x end exponent plus e to the power of negative x end exponent close parentheses plus 1 equals 0 is given by

    Maths-General

    When we get multipliction of two brackets equal to zero, any of the bracket can be zero or both of the brackets can be zero. It is decided based on the conditions in the equation.

    parallel
    General
    Maths-

    The differential equation of all ellipse centred at the origin and major and minor axes along coordinate axes is

    The differential equation of all ellipse centred at the origin and major and minor axes along coordinate axes is

    Maths-General
    General
    Maths-

    Area of the region bounded by y equals tan invisible function application x comma tangent drawn to the curve at x equals fraction numerator pi over denominator 4 end fraction and the x minusaxis is

    Area of the region bounded by y equals tan invisible function application x comma tangent drawn to the curve at x equals fraction numerator pi over denominator 4 end fraction and the x minusaxis is

    Maths-General
    General
    Maths-

    Maximum value of x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent, when 0 less or equal than x less or equal than 2, is

    Maximum value of x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent, when 0 less or equal than x less or equal than 2, is

    Maths-General
    parallel
    General
    Maths-

    If x = c o s to the power of negative 1 end exponentt, y = square root of 1 minus t to the power of 2 end exponent end root , then y subscript 2 end subscript equals

    If x = c o s to the power of negative 1 end exponentt, y = square root of 1 minus t to the power of 2 end exponent end root , then y subscript 2 end subscript equals

    Maths-General
    General
    Physics-

    Two capacitors C subscript 1 end subscript equals 2 mu F and C subscript 2 end subscript equals 6 mu F in series, are connected in parallel to a third capacitor C subscript 3 end subscript equals 4 mu F. This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

    Two capacitors C subscript 1 end subscript equals 2 mu F and C subscript 2 end subscript equals 6 mu F in series, are connected in parallel to a third capacitor C subscript 3 end subscript equals 4 mu F. This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

    Physics-General
    General
    Physics-

    The combination of capacitors with C subscript 1 end subscript equals 3 mu F comma C subscript 2 end subscript equals 4 mu F and C subscript 3 end subscript equals 2 mu F is charged by connecting AB to a battery. Consider the following statements
    I. Energy stored in C subscript 1 end subscript= Energy stored in C subscript 2 end subscript + Energy stored in C subscript 3 end subscript
    II. Charge on C1 = Charge on C2 + Charge on C3
    III. Potential drop across C1 = Potential drop across C2 = Potential drop across C3
    Which of these is/are correct

    The combination of capacitors with C subscript 1 end subscript equals 3 mu F comma C subscript 2 end subscript equals 4 mu F and C subscript 3 end subscript equals 2 mu F is charged by connecting AB to a battery. Consider the following statements
    I. Energy stored in C subscript 1 end subscript= Energy stored in C subscript 2 end subscript + Energy stored in C subscript 3 end subscript
    II. Charge on C1 = Charge on C2 + Charge on C3
    III. Potential drop across C1 = Potential drop across C2 = Potential drop across C3
    Which of these is/are correct

    Physics-General
    parallel
    General
    Physics-

    Consider a parallel plate capacitor of 10 mu rightwards arrow over short leftwards arrow F (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

    Consider a parallel plate capacitor of 10 mu rightwards arrow over short leftwards arrow F (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

    Physics-General
    General
    Physics-

    In the figure a capacitor is filled with dielectrics. The resultant capacitance is

    In the figure a capacitor is filled with dielectrics. The resultant capacitance is

    Physics-General
    General
    Physics-

    Equivalent capacitance between A and B is

    Equivalent capacitance between A and B is

    Physics-General
    parallel

    card img

    With Turito Academy.

    card img

    With Turito Foundation.

    card img

    Get an Expert Advice From Turito.

    Turito Academy

    card img

    With Turito Academy.

    Test Prep

    card img

    With Turito Foundation.