Question
Let a matrix A = 
& P =
Q = PAPT where PT is transpose of matrix P. Find PT Q2005 P is
Hint:
check whether inverse of P exists or not and then use the properties of invertible matrices while solving the expression.
The correct answer is:
option A
A2005 = [ 1 2005
0 1]
PTP = (1 0
0 1)
=> PTP = I or PT = P-1
Now, evaluating the required expression,
PT Q 2005 P = PT [(PAPT) (PAPT) (PAPT) (PAPT) (PAPT)….]P
Since P is invertible, we can write
(IA)(IA)(IA)….2005 times
Or, I A2005 = A2005
Now, calculating A2005 ,
A = [1 1
0 1]
A2 = [1 2
0 1]
A3 = [1 3
0 1]
Similarly, A2005 = [ 1 2005
0 1]
an invertible matrix gives identity matrix as the product when multiplied with its inverse.
Related Questions to study
Let A =
& I =
and A-1= 
[A2+CA+dI], find ordered pair (c,d) ?
the roots of the characteristic equation give us the eigenvalues of the matrix.
Let A =& I = and A-1= [A2+CA+dI], find ordered pair (c,d) ?
the roots of the characteristic equation give us the eigenvalues of the matrix.
Which of the following is an intermolecular redox reaction?
Which of the following is an intermolecular redox reaction?
In which of the following processes is nitrogen oxidised?
In which of the following processes is nitrogen oxidised?
Which one of the following statements, is true-
A square matrix is one whose number of rows are equal to the number of columns.
Which one of the following statements, is true-
A square matrix is one whose number of rows are equal to the number of columns.
If A = 
and A adj A = 
, then k is equal to -
adjoint of a matrix is the transpose of the cofactor matrix of that matrix. cofactors are the resultant numbers when the corresponding row and column of an element is removed from the matrix.
If A = and A adj A = , then k is equal to -
adjoint of a matrix is the transpose of the cofactor matrix of that matrix. cofactors are the resultant numbers when the corresponding row and column of an element is removed from the matrix.
If Aαβ=, then which of following statement is true -
If Aαβ=, then which of following statement is true -
If 
then
If then
Which of following is not a disproportionation reaction?
Which of following is not a disproportionation reaction?
Consider a continuous function 
for all a, b in the domain of ‘f’ and 
is a non zero finite number then
Consider a continuous function for all a, b in the domain of ‘f’ and is a non zero finite number then
The figure shows four pairs of polarizing sheets, seen face-on. Each pair is mounted in the path of initially unpolarized light. The polarizing direction of each sheet (indicated by the dashed line) is referenced to either a horizontal x-axis or a vertical y axis. Rank the pair according to the fraction of the initial intensity that they pass, greatest first
The figure shows four pairs of polarizing sheets, seen face-on. Each pair is mounted in the path of initially unpolarized light. The polarizing direction of each sheet (indicated by the dashed line) is referenced to either a horizontal x-axis or a vertical y axis. Rank the pair according to the fraction of the initial intensity that they pass, greatest first
The figure here gives the electric field of an EM wave at a certain point and a certain instant. The wave is transporting energy in the negative z direction. What is the direction of the magnetic field of the wave at that point and instant
The figure here gives the electric field of an EM wave at a certain point and a certain instant. The wave is transporting energy in the negative z direction. What is the direction of the magnetic field of the wave at that point and instant
In the set up shown in Fig the two slits, 
and 
are not equidistant from the slit S. The central fringe at O is then
In the set up shown in Fig the two slits, and are not equidistant from the slit S. The central fringe at O is then
In a Young’s double slit experimental arrangement shown here, if a mica sheet of thickness t and refractive index 
is placed in front of the slit 
then the path difference 
In a Young’s double slit experimental arrangement shown here, if a mica sheet of thickness t and refractive index is placed in front of the slit then the path difference
The compound (A) is :
The compound (A) is :
A) Requires 2 mol of 
for its saturation
B) Reduces ammoniacal 
and gives yellow colour with 
C) does not reduce Tollens reagent but gives iodoform test
D) n dehydration with conc. 
gives a mixture of colourless gases
The compound (A) is:
A) Requires 2 mol of for its saturation
B) Reduces ammoniacal and gives yellow colour with
C) does not reduce Tollens reagent but gives iodoform test
D) n dehydration with conc. gives a mixture of colourless gases
The compound (A) is:
