If A- 2 1- 4 1 - B- 3 4- 2 3 -and C- 3 -4- -2 3 - then trA-trABC-2 -trABC2-4-trABC3-8 - -

The correct option is C 6 B=[ 3 amp; 4 2 amp; 3 ] #160; ; #160; C=[ 3 amp; 4 2 amp; 3 ] ⇒ BC=[ 1 ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

If A=[ 2 1;...

Question

If $A = [\begin{matrix} 2 & 1 4 & 1 \end{matrix}]$ , $B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}]$ and $C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}]$ , then $t r (A) + t r (\frac{A B C}{2}) + t r (\frac{A {(B C)}^{2}}{4}) + t r (\frac{A {(B C)}^{3}}{8}) + . . . . . . . . + \infty =$

6

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Solution

The correct option is C $6$
$B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}]$ $;$ $C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}]$
$\Rightarrow B C = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]$ $\Rightarrow B C = I$
$∴ A B C = A . I = A$
$A {(B C)}^{2} = A I^{2} = A$
$∴ t r (A) + t r (\frac{A B C}{2}) + t r . (\frac{A {(B C)}^{2}}{4}) + t r . (\frac{A {(B C)}^{3}}{8}) + . . . . . . . .$
$= 3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + . . . . . . + A$
$= 3 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{1 - \frac{1}{2}} ⎞ ⎟ ⎟ ⎟ ⎠$
$= \frac{3}{\frac{1}{2}}$

$= 6$
Hence, the answer is $6.$

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Similar questions

Q. Let three matrices A =

[\begin{matrix} 2 & 1 4 & 1 \end{matrix}]; B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}] a n d C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}]

then

t_{r} (A) + t_{r} (\frac{A B C}{2}) + t_{r} (\frac{A (B C)^{2}}{4}) + t_{r} (\frac{A (B C)^{3}}{8}) + . . . . + \infty

Q. If

A = [\begin{matrix} 2 & 1 4 & 1 \end{matrix}], B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}] and C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}],

then the value of

t r (A) + t r (\frac{A B C}{2}) + t r (\frac{A (B C)^{2}}{4}) + t r (\frac{A (B C)^{3}}{8}) + . . . \infty

Q. Consider three matrices

A = [\begin{matrix} 2 & 1 4 & 1 \end{matrix}] B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}]

and

C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}]

.
The value of the sum

t r (A) + t r (\frac{A B C}{2}) + t r (\frac{A ({B C)}^{2}}{4}) + t r (\frac{A ({B C)}^{3}}{8}) + . . . . . . . . . . . . . + \infty

is
(

t r (A)

denotes trace of a matrix

A

)

Q. Let three matrices

A = [\begin{matrix} 2 & 1 4 & 1 \end{matrix}]

;

B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}]

and

C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}]

then find

t r (A) + t r (\frac{A B C}{2}) t r (\frac{A {(B C)}^{2}}{4}) + t r (\frac{A {(B C)}^{3}}{8}) + . . . . + \infty

, where

t r (A)

represents trace of matrix

A

If A = {3, 6, 9, 12, 15, 18, 21}
B = {2, 4, 6, 8, 10, 12, 14, 16}
C = {5, 10, 15, 20}

$\begin{matrix} A & B 1. B - C & A . {5, 10, 20} 2. A - C & B . {3, 9, 15, 21} 3. C - (A - B) & C . {3, 6, 9, 12, 18, 21} 4. C \cap (A - B) & D . {15} E . {2, 4, 6, 8, 12, 14, 16} \end{matrix}$

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If A=[2141], B=[3423]and C=[3−4−23], then tr(A)+tr(ABC2)+tr(A(BC)24)+tr(A(BC)38)+........+∞=

The correct option is C 6B=[3423] ; C=[3−4−23]⇒BC=[1001] ⇒BC=I∴ABC=A.I=A A(BC)2=AI2=A∴tr(A)+tr(ABC2)+tr.(A(BC)24)+tr.(A(BC)38)+........=3+32+34+38+......+A=3⎛⎜ ⎜ ⎜⎝11−12⎞⎟ ⎟ ⎟⎠=312=6Hence, the answer is 6.

If $A = [\begin{matrix} 2 & 1 4 & 1 \end{matrix}]$ , $B = [\begin{matrix} 3 & 4 2 & 3 \end{matrix}]$ and $C = [\begin{matrix} 3 & - 4 - 2 & 3 \end{matrix}]$ , then $t r (A) + t r (\frac{A B C}{2}) + t r (\frac{A {(B C)}^{2}}{4}) + t r (\frac{A {(B C)}^{3}}{8}) + . . . . . . . . + \infty =$