If A- 0 - 0 0 - and A-I50-50A- a b- c d - then the value of a-b-c-d is

Name: JEE- Grade 12- Mathematics- Matrices and Determinants- Session 03- D09
Uploaded: 2022-07-04T10:36:42+05:30
Description: We have, A^2=[ 0 amp;α; nbsp; nbsp;0 amp;0 nbsp; ][ 0 amp;α; nbsp; nbsp;0 amp;0 nbsp; ] =[ 0 amp;0; nbsp; nbsp;0 amp;0 nbsp; ]=O ⇒ ...

We have, A^2=[ 0 amp;α; nbsp; nbsp;0 amp;0 nbsp; ][ 0 amp;α; nbsp; nbsp;0 amp;0 nbsp; ] =[ 0 amp;0; nbsp; nbsp;0 amp;0 nbsp; ]=O ⇒ ...

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Byju's Answer

Standard X

Mathematics

Equality of Matrices

If A=[ 0 α; 0...

Question

If $A = [\begin{matrix} 0 & α 0 & 0 \end{matrix}]$ and $(A + I)^{50} - 50 A = [\begin{matrix} a & b c & d \end{matrix}],$ then the value of $a + b + c + d$ is

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Solution

We have, $A^{2} = [\begin{matrix} 0 & α 0 & 0 \end{matrix}] [\begin{matrix} 0 & α 0 & 0 \end{matrix}]$
$= [\begin{matrix} 0 & 0 0 & 0 \end{matrix}] = O$
$\Rightarrow A^{k} = O, \forall k \geq 2$
Thus,
$(A + I)^{2} = A^{2} + I^{2} + A I + I A = I + 2 A$
$(A + I)^{3} = (A + I)^{2} (A + I) = (I + 2 A) (A + I) = I + 3 A \cdot \cdot (A + I)^{50} = I + 50 A \Rightarrow (A + I)^{50} - 50 A = I + 50 A - 50 A = I \Rightarrow [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] = [\begin{matrix} a & b c & d \end{matrix}] \Rightarrow a = 1, b = 0, c = 0, d = 1 \Rightarrow a + b + c + d = 2$

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Similar questions

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

u_{1}

and

u_{2}

are column matrices such that

A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦

and

A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦

, then

u_{1} + u_{2}

is equal to

$L e t A = [\begin{matrix} 0 & α 0 & 0 \end{matrix}] a n d (A + I)^{50} - 50 A = [\begin{matrix} a & b c & d \end{matrix}], T h e n, t h e v a l u e o f a + b + c + d i s$