For x-0- let A - - x-1x 0 0- 0 x 0- 0 0 16 - and B - - 5xx2-1 0 0- 0 3x 0- 0 0 14 - be two matrices-Three other matrices X- Y and Z are defined asX - AB-1 - AB-2 - AB-3 - - AB-n- Y - limn -X and Z - Y-1-2I- where I is the identity matrix of order 3- If trP denotes the trace of matrix P and adj P denotes the adjoint of matrix P- then which of the following is-are CORRECT-

AB = [ x^2+1x amp; 0 amp; 0; nbsp; 0 amp; x amp;0; nbsp; 0 amp; 0 amp;16 nbsp; ][ 5xx^2+1 amp; 0 amp; 0; nbsp; 0 amp;3x nbsp; amp; 0; nbsp; 0 a ...

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Byju's Answer

Standard XI

Mathematics

Multiplication of Matrices

For x>0, let ...

Question

For $x > 0,$ let $A = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x + \frac{1}{x} & 0 & 0 0 & x & 0 0 & 0 & 16 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$ and $B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{5 x}{x^{2} + 1} & 0 & 0 0 & \frac{3}{x} & 0 0 & 0 & \frac{1}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$ be two matrices.
Three other matrices $X, Y$ and $Z$ are defined as
$X = (A B)^{- 1} + (A B)^{- 2} + {A B}^{- 3} + \dots (A B)^{- n},$ $Y = lim n \to \infty X$ and $Z = Y^{- 1} - 2 I,$ where $I$ is the identity matrix of order $3.$ If $t r (P)$ denotes the trace of matrix $P$ and $a d j (P)$ denotes the adjoint of matrix $P$ , then which of the following is/are CORRECT?

∣ ∣ a d j (\sqrt{5} Y^{- 1}) ∣ ∣ = 5 (5!)^{2}

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∣ ∣ a d j (\sqrt{5} Y^{- 1}) ∣ ∣ = (5!)^{2}

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The least positive integral value of

t r (A Y)

7

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t r (Z + Z^{2} + Z^{3} + \dots + Z^{10}) = 2^{11} + 8

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Solution

The correct option is D $t r (Z + Z^{2} + Z^{3} + \dots + Z^{10}) = 2^{11} + 8$
$A B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{x^{2} + 1}{x} & 0 & 0 0 & x & 0 0 & 0 & 16 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{5 x}{x^{2} + 1} & 0 & 0 0 & \frac{3}{x} & 0 0 & 0 & \frac{1}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ \Rightarrow A B = ⎡ ⎢ ⎣ \begin{matrix} 5 & 0 & 0 0 & 3 & 0 0 & 0 & 4 \end{matrix} ⎤ ⎥ ⎦$

$(A B)^{- 1} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{5} & 0 & 0 0 & \frac{1}{3} & 0 0 & 0 & \frac{1}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

$(A B)^{- 2} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{5^{2}} & 0 & 0 0 & \frac{1}{3^{2}} & 0 0 & 0 & \frac{1}{4^{2}} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
and so on.

$∴ X = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{5} + \frac{1}{5^{2}} + \dots + \frac{1}{5^{n}} & 0 & 0 0 & \frac{1}{3} + \frac{1}{3^{2}} + \dots + \frac{1}{3^{n}} & 0 0 & 0 & \frac{1}{4} + \frac{1}{4^{2}} + \dots + \frac{1}{4^{n}} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Y = lim n \to \infty X = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{\frac{1}{5}}{1 - \frac{1}{5}} & 0 & 0 0 & \frac{\frac{1}{3}}{1 - \frac{1}{3}} & 0 0 & 0 & \frac{\frac{1}{4}}{1 - \frac{1}{4}} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ \Rightarrow Y = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{4} & 0 & 0 0 & \frac{1}{2} & 0 0 & 0 & \frac{1}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∴ Y^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 4 & 0 & 0 0 & 2 & 0 0 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦$

$∣ ∣ a d j (\sqrt{5} Y^{- 1}) ∣ ∣ = ∣ ∣ (\sqrt{5})^{2} a d j (Y^{- 1}) ∣ ∣ = 5^{3} \cdot | Y^{- 1} |^{2} = 5^{3} 24^{2} = 5 \cdot (5!)^{2}$

$A Y = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x + \frac{1}{x} & 0 & 0 0 & x & 0 0 & 0 & 16 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{4} & 0 & 0 0 & \frac{1}{2} & 0 0 & 0 & \frac{1}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ \Rightarrow A Y = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{4} (x + \frac{1}{x}) & 0 & 0 0 & \frac{x}{2} & 0 0 & 0 & \frac{16}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ \Rightarrow t r (A Y) = \frac{x}{4} + \frac{1}{4 x} + \frac{x}{2} + \frac{16}{3} \Rightarrow t r (A Y) = \frac{3 x}{4} + \frac{1}{4 x} + \frac{16}{3}$
As $x > 0$ , using $A . M . \geq G . M .$ , we get
$\frac{3 x}{4} + \frac{1}{4 x} \geq 2 \sqrt{\frac{3}{16}} \Rightarrow \frac{3 x}{4} + \frac{1}{4 x} \geq \frac{\sqrt{3}}{2}$

So, $t r (A Y) \geq \frac{\sqrt{3}}{2} + \frac{16}{3}$
$∴$ Least positive integral value of $t r (A Y) = 7$

$Z = Y^{- 1} - 2 I = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 0 0 & 0 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ \Rightarrow Z^{n} = ⎡ ⎢ ⎣ \begin{matrix} 2^{n} & 0 & 0 0 & 0 & 0 0 & 0 & 1^{n} \end{matrix} ⎤ ⎥ ⎦$

$t r (Z + Z^{2} + Z^{3} + \dots + Z^{10}) = 2 + 2^{2} + 2^{3} + \dots + 2^{10} + 10 = 2 \cdot (\frac{2^{10} - 1}{2 - 1}) + 10 = 2^{11} + 8$

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Similar questions

Q. For

x > 0,

let

A = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x + \frac{1}{x} & 0 & 0 0 & x & 0 0 & 0 & 16 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

and

B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{5 x}{x^{2} + 1} & 0 & 0 0 & \frac{3}{x} & 0 0 & 0 & \frac{1}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

be two matrices.
Two other matrices

X

and

Y

are defined as

X = (A B)^{- 1} + (A B)^{- 2} + {A B}^{- 3} + \dots (A B)^{- n}

and

Y = lim n \to \infty X

. If

t r (P)

denotes the trace of matrix

P

, then the least positive integral value of

t r (A Y)

Q. For

x > 0,

let

A = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x + \frac{1}{x} & 0 & 0 0 & x & 0 0 & 0 & 16 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

and

B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{5 x}{x^{2} + 1} & 0 & 0 0 & \frac{3}{x} & 0 0 & 0 & \frac{1}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

be two matrices.
Three other matrices

X, Y

and

Z

are defined as

X = (A B)^{- 1} + (A B)^{- 2} + {A B}^{- 3} + \dots (A B)^{- n},

Y = lim n \to \infty X

and

Z = Y^{- 1} - 2 I,

where

I

is the identity matrix of order

3.

t r (P)

denotes the trace of matrix

P

and

a d j (P)

denotes the adjoint of matrix

P

, then which of the following is/are CORRECT?

Q. For x > 0, Let

A = ⎡ ⎢ ⎣ \begin{matrix} x + \frac{1}{x} & 0 & 0 0 & x & 0 0 & 0 & 16 \end{matrix} ⎤ ⎥ ⎦ B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{5 x}{x^{2} + 1} & 0 & 0 0 & \frac{3}{x} & 0 0 & 0 & \frac{1}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

X = (A B)^{- 1} + (A B)^{- 2} + (A B)^{- 3} + . . . \infty

Z = X^{- 1} - 2 I

(I is identity matrix of order 3)

\begin{matrix} (P) minimum value of [Tr(Ax)] i s & (1) 24 (when [.]) \to represent integer function (Q) det (X^{- 1}) i s & (2) 12 (R) If T r (z + z^{2} + - - - + z^{10}) = 2^{a} + b, (a, b \in N) then a + b is & (3) 6 (S) If value of | a d j (\sqrt{5} X^{- 1}) | = k then number of positive divisors & (4) 19 of k which are odd is \end{matrix}

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 7 0 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} - x & 14 x & 7 x 0 & 1 & 0 x & - 4 x & - 2 x \end{matrix} ⎤ ⎥ ⎦

are two matrices such that

A B = (A B)^{- 1}

and

A B \neq I

(where

I

is an identity matrix of order

3 \times 3

).
Find the value of

T r . (A B + (A B)^{2} + (A B)^{3} + . . . + (A B)^{100})

where

T r . (A)

denotes the trace of matrix

A

Q. Let

A = ⎛ ⎜ ⎝ \begin{matrix} x^{2} & 6 & 8 3 & y^{2} & 9 4 & 5 & z^{2} \end{matrix} ⎞ ⎟ ⎠

and

B = ⎛ ⎜ ⎝ \begin{matrix} 2 x & 3 & 5 2 & 2 y & 6 1 & 4 & 2 z - 3 \end{matrix} ⎞ ⎟ ⎠

be two matrices and if

T r (A) = T r (B)

, then the value of

(x + y + z)

is equal to
(Note:

T r (P)

denotes trace of matrix

P

)

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