Home
Class 12
MATHS
Let p=[(3,-1,-2),(2,0,alpha),(3,-5,0)], ...

Let `p=[(3,-1,-2),(2,0,alpha),(3,-5,0)],` where `alpha in RR.` Suppose `Q=[q_(ij)]` is a matrix such that `PQ=kI,` where `k in RR, k != 0 and I`is the identity matrix of order 3. If `q_23=-k/8 and det(Q)=k^2/2,` then

A

`alpha = 0, k=8`

B

`4alpha -k + 8 =0`

C

`det (padj(Q) ) = 2^(9)`

D

`det (Qadj(P) ) = 2^(13)`

Text Solution

Verified by Experts

The correct Answer is:
B, C
`because PQ = kI rArr (P.Q)/k = I rArr P^(-1) = Q/k" " ...(i)`
Also `abs(P) = 12 alpha +20 " " (ii)`
and `" "`given ` q_(23) = (-k)/8`
Comoaring the third element of `2^(nd)` row no both sides,
we get `1/((12alpha + 20))(-(3alpha + 4)) = 1/k xx (-k)/8`
`rArr 24 alpha + 32= 12 alpha + 20`
` alpha = -1` ...(iii)
From (ii), `abs(P)=8` ...(iv)
Also `PQ= kI`
`rArrabs(PQ) = abs(kI)`
`rArr abs(P) abs(Q) = k^(3)`
`rArr 8xx (k^(2))/2 = k^(3) " " (because abs(P) = 8, abs(Q) = k^(2)/2)`
`therefore k= 4 " " ...(v)`
(b) `4 alpha - k + 8 = -4 -4 + 8 = 0`
(c) `det(P " adj" (Q)) = abs(P) abs("adj" Q) = abs(P) abs(Q)^(2) = 8xx8^(2) = 2^(9)`
(d) `det(Q " adj" (P)) = abs(Q) abs("adj" P) = abs(Q) abs(P)^(2) = 8xx8^(2) = 2^(9)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If M is a 3xx3 matrix, where det M=1a n dM M^T=I,w h e r eI is an identity matrix, prove that det (M-I)=0.

Let A=[(0,1),(0,0) ] show that (a I+b A)^n=a^n I+n a^(n-1)b A , where I is the identity matrix of order 2 and n in N .

Let A=[(0,1),(0,0)] , show that (aI+bA)^(n)=a^(n)I+na^(n-1)bA , where I is the identity matrix of order 2 and n in N .

If A is a square matrix of order 3 and I is an ldentity matrix of order 3 such that A^(3) - 2A^(2) - A + 2I =0, then A is equal to

If A =[{:(3,-3,4),(2,-3,4),(0,-1,1):}] and B is the adjoint of A, find the value of |AB+2I| ,where l is the identity matrix of order 3.

Let P=[(1,0,0),(3,1,0),(9,3,1)] and Q = [q_(ij)] be two 3xx3 matrices such that Q - P^(5) = I_(3) . Then (q_(21)+q_(31))/(q_(32)) is equal to

If A= Verify A^3-3A^2-10A+24I=0 where O is zero matrix of order 3xx3

If A is an idempotent matrix satisfying (I-0.4A)^(-1)=I-alphaA where I is the unit matrix of the same order as that of A then the value of alpha is

If P = (0,1,2) and Q = (4,-2,1), O = (0,0,0) then P hat O Q =

If P is a 3xx3 matrix such that P^(T) = 2P +I, where P^(T) is the transpose f P and I is the 3xx3 identify matrix then there exists a column matrix X = [(x),(y),(z)] != [ (0),(0),(0)] such that :