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Maths-

General

Easy

Question

If A = $open square brackets table row 1 k row 0 1 end table close square brackets$ then Aⁿequal to-

$[\begin{matrix} 1 & k^{n} \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} 1 & n k \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} k^{n} & 1 \\ 0 & 1 \end{matrix}]$
None of these

hint Hint:

To multiply one matrix with another, we need to check first, if the number of columns of the first matrix is equal to the number of rows of the second matrix. Now multiply each element of the column of the first matrix with each element of rows of the second matrix and add them all.

The correct answer is: $open square brackets table row 1 cell n k end cell row 0 1 end table close square brackets$

$D e t a i l e d space S o l u t i o n G i v e n space A to the power of 1 equals space open square brackets table row cell 1 space space k end cell row cell 0 space space 1 end cell end table close square brackets space comma space A to the power of n equals ? space space space space space space space n element of N space space space space left parenthesis space n equals 1 comma 2 comma 3 comma 4 horizontal ellipsis.. right parenthesis$
Using Multiplication of Matrices ,

$A squared equals space A. A space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space A squared equals space open square brackets table row cell 1 space space k end cell row cell 0 space space 1 end cell end table close square brackets open square brackets table row cell 1 space space k end cell row cell 0 space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space A squared space space equals space open square brackets table row cell 1 plus 0 space space space space k plus k end cell row cell 0 plus 0 space space space 0 plus 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space A squared equals space open square brackets table row cell 1 space space space space 2 k end cell row cell 0 space space space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space S i m i l a r l y comma space A cubed equals space A squared. A A cubed equals space open square brackets table row cell 1 space space space space 2 k end cell row cell 0 space space space space 1 end cell end table close square brackets space open square brackets table row cell 1 space space space k end cell row cell 0 space space space 1 end cell end table close square brackets space A cubed space equals open square brackets table row cell 1 plus 0 space space space space space k plus 2 k end cell row cell 0 plus 0 space space space space space space 0 plus 1 end cell end table close square brackets A cubed space equals open square brackets table row cell 1 space space space space 3 k end cell row cell 0 space space space space 1 space space end cell end table close square brackets S i m i l a r l y space A to the power of 4 equals space A cubed. A A to the power of 4 space equals space open square brackets table row cell 1 space space space 3 k end cell row cell 0 space space space space 1 end cell end table close square brackets open square brackets table row cell 1 space space space k end cell row cell 0 space space space space 1 end cell end table close square brackets space A to the power of 4 space equals open square brackets table row cell 1 plus 0 space space space space space space k plus 3 k end cell row cell 0 plus 0 space space space space space space 0 plus 1 end cell end table close square brackets A to the power of 4 equals space open square brackets table row cell 1 space space space 4 k end cell row cell 0 space space space 1 end cell end table close square brackets$
By observing the pattern,
$A to the power of 1 comma space A squared comma space A cubed comma space A to the power of 4 comma space w e space c a n space s a y A to the power of n equals space open square brackets table row cell 1 space space n k end cell row cell 0 space space 1 end cell end table close square brackets$

Related Questions to study

If A and B are two matrix of same order then A² – B² = (A + B) (A – B) correct statement is-

If A and B are two matrix of same order then A² – B² = (A + B) (A – B) correct statement is-

If A = $open square brackets table row 1 1 row 0 1 end table close square brackets$ then Aⁿ =

If A = $open square brackets table row 1 1 row 0 1 end table close square brackets$ then Aⁿ =

If A = $open square brackets table row 1 alpha row 0 1 end table close square brackets$ , then A⁴ is equal to-

If A = $open square brackets table row 1 alpha row 0 1 end table close square brackets$ , then A⁴ is equal to-

parallel

If A = $open square brackets table row 0 1 row 0 0 end table close square brackets$ , I is the unit matrix of the order two & a, b are arbitrary constant then
(aI + bA)²=

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(aI + bA)²=

If AB = C; then A, B, C are-

If AB = C; then A, B, C are-

The magnitude of I in ampere is

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physics-General

parallel

A cord of length 64 m is used to connect a 100 kg astronaut to a space-ship whose mass is much larger than that of the astronaut. Estimate the value of the tension in the cord. Assume that the space-ship is orbiting near earth surface. Also assume that the space-ship and the astronaut fall on a straight line from the earth’s centre. The radius of the earth is 6400 km.

A cord of length 64 m is used to connect a 100 kg astronaut to a space-ship whose mass is much larger than that of the astronaut. Estimate the value of the tension in the cord. Assume that the space-ship is orbiting near earth surface. Also assume that the space-ship and the astronaut fall on a straight line from the earth’s centre. The radius of the earth is 6400 km.

physics-General

If for a matrix A, A³ = I then A^–1 equals -

If for a matrix A, A³ = I then A^–1 equals -

The temperature of the two outer surfaces of a composite slab, consists of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x respectively $text are end text T subscript 2 end subscript text end text$ and $text end text T subscript 1 end subscript open parentheses T subscript 2 end subscript greater than T subscript 1 end subscript close parentheses$
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The temperature of the two outer surfaces of a composite slab, consists of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x respectively $text are end text T subscript 2 end subscript text end text$ and $text end text T subscript 1 end subscript open parentheses T subscript 2 end subscript greater than T subscript 1 end subscript close parentheses$
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physics-General

parallel

Three rods of material x and three rods of material y are connected as shown in figure all the rods are of identical length and cross section If the end A is maintained at 60 $degree$ C and the junction E at 10 $degree$ C, find effective Thermal Resistance Given length of each rod=l, area of cross-section=A, conductivity of x=K and conductivity of y=2K

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physics-General

One end of a copper rod of uniform cross section and of length 1.5m is kept in contact with ice and the other end with water at 100 $degree$ C At what point along its length should a temperature of 200 $degree$ C be maintained so that in steady sate, the mass of ice melting be equal to that of the steam produced in same interval of time? Assume that the whole system is insulated from surroundings $open parentheses L subscript text ice end text end subscript equals 80 c a l divided by g m close text end text$ and $text end text open L subscript text steam end text end subscript equals 540 c a l divided by g m close parentheses$

One end of a copper rod of uniform cross section and of length 1.5m is kept in contact with ice and the other end with water at 100 $degree$ C At what point along its length should a temperature of 200 $degree$ C be maintained so that in steady sate, the mass of ice melting be equal to that of the steam produced in same interval of time? Assume that the whole system is insulated from surroundings $open parentheses L subscript text ice end text end subscript equals 80 c a l divided by g m close text end text$ and $text end text open L subscript text steam end text end subscript equals 540 c a l divided by g m close parentheses$

physics-General

If A = $open square brackets table row 1 2 3 row 2 3 4 row 0 0 2 end table close square brackets$ , then the value of adj (adj A) is -

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parallel

The curves for potential energy (U) and kinetic energy $left parenthesis E subscript k end subscript right parenthesis$ of a two particle system are shown in figure. At what points the system will be bound?

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physics-General

Suppose, the acceleration due to gravity at the earth’s surface is 10 m/s² and at the surface of Mars it is 4.0 m/s². A 60 kg passenger goes from the earth to the Mars in a spaceship moving with a constant velocity. Neglect all other objects in the sky. Which part of figure best represents the weight (net gravitational force)of the passenger as a function of time.

Suppose, the acceleration due to gravity at the earth’s surface is 10 m/s² and at the surface of Mars it is 4.0 m/s². A 60 kg passenger goes from the earth to the Mars in a spaceship moving with a constant velocity. Neglect all other objects in the sky. Which part of figure best represents the weight (net gravitational force)of the passenger as a function of time.

physics-General

A sphere of mass M and radius R₂ has a concentric cavity of radius R₁ as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as $left parenthesis 0 less or equal than r less or equal than infinity right parenthesis$

A sphere of mass M and radius R₂ has a concentric cavity of radius R₁ as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as $left parenthesis 0 less or equal than r less or equal than infinity right parenthesis$

physics-General

parallel

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