Let A-2 0 7 0 1 0 1 -2 1- and B-x 14x 7x 0 1 0 x -4x -2x- are two matrices such that AB - AB-1 and AB I where I is an identity matrix of order 3-3-Find the value of Tr-AB-AB2-AB3-AB100 where Tr-A denotes the trace of matrix A-

The correct option is C 100 nbsp; A=[2 amp;0 amp;7 0 amp;1 amp;0 1 amp; 2 amp;1] and B=[ x amp;14x amp;7x 0 amp;1 a ...

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Byju's Answer

Standard XII

Mathematics

Identity Matrix

Let A=[2 0...

Question

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 $A$ .

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Solution

The correct option is C 100
$A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 7 0 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} - x & 14 x & 7 x 0 & 1 & 0 x & - 4 x & - 2 x \end{matrix} ⎤ ⎥ ⎦$
$A B = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 7 0 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} - x & 14 x & 7 x 0 & 1 & 0 x & - 4 x & - 2 x \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 5 x & 14 x & 0 0 & 1 & 0 0 & 10 x - 2 & 5 x \end{matrix} ⎤ ⎥ ⎦$
but, $A B = (A B)^{- 1} \Rightarrow (A B)^{2} = I$
$\Rightarrow (A B)^{2} = ⎡ ⎢ ⎣ \begin{matrix} 5 x & 14 x & 0 0 & 1 & 0 0 & 10 x - 2 & 5 x \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 5 x & 14 x & 0 0 & 1 & 0 0 & 10 x - 2 & 5 x \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 25 x^{2} & 70 x^{2} + 14 x & 0 0 & 1 & 0 0 & (5 x + 1) (10 x - 2) & 25 x^{2} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow x = \frac{- 1}{5}$
$T r . (A B + (A B)^{2} + (A B)^{3} + . . . + (A B)^{100}) = T r . (A B + I + (A B) + . . . + I)$
$t r (A B) = 10 x + 1 = - 1$ and $t r (I) = 3$
$∴ T r . (A B + I + (A B) + . . . + I) = (- 1 + 3 - 1 + 3...... + 3) = 50 (3 - 1) = 100$
Hence, option C.

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Similar questions

Q. Find the value of x for which the product of the follwing matrices equals an identity matrix.

⎛ ⎜ ⎝ \begin{matrix} 2 & 0 & 7 0 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ \times ⎛ ⎜ ⎝ \begin{matrix} - x & 14 x & 7 x 0 & 1 & 0 x & - 4 x & - 2 x \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} ⎤ ⎥ ⎦

be 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.

Then the value of

t r (A B + (A B)^{2} + (A B)^{3} + \dots + (A B)^{100})

(

Here,

t r (A)

denotes the trace of matrix

A,

i.e., sum of diagonal elements of

A .)

Q. Let

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

and

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

. If

A B = I

, where

I

is an identity matrix of order

3

then trace

B

has value equal to

Q. Let

A

and

B

3 \times 3

square matrices such that

A B = 9 I

where

A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & - 1 4 & - 3 & 4 3 & - 3 & λ \end{matrix} ⎤ ⎥ ⎦

b_{33} = 9. a_{23}

, then value of

tr (A^{2} + B^{2})

is
(

tr

denotes trace of the matrix)

Q. Let

A = ⎡ ⎣ \begin{matrix} 1 & \frac{3}{2} 1 & 2 \end{matrix} ⎤ ⎦, B = [\begin{matrix} 4 & - 3 - 2 & 2 \end{matrix}] and C_{r} = [\begin{matrix} r {.3}^{r} & 2^{r} 0 & (r - 1) 3^{r} \end{matrix}]

be 3 given matrices. Compute the value of

\sum_{r = 1}^{50} t r . ((A B)^{r} C_{r}) . (

where

t r . (A)

denotes trace of matrix A

)

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Let A=⎡⎢⎣2070101−21⎤⎥⎦ and B=⎡⎢⎣−x14x7x010x−4x−2x⎤⎥⎦ are two matrices such that AB=(AB)−1 and AB≠I (where I is an identity matrix of order 3×3).Find the value of Tr.(AB+(AB)2+(AB)3+...+(AB)100) where Tr.(A) denotes the trace of matrix A.