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Matrices and Determinants question

2019 · 12 Jan · Shift 1 · Q33
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  5. /2019 · 12 Jan · Shift 1 · Q33

Matrices and Determinants question

2019 · 12 Jan · Shift 1 · Q33

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
Let P = [100310931]\left[ {\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \\ \end{matrix} } \right]​139​013​001​​ and Q = [qij] be two 3 ×\times× 3 matrices such that Q – P5 = I3. Then q21+q31q32{{{q_{21}} + {q_{31}}} \over {{q_{32}}}}q32​q21​+q31​​ is equal to :
  1. A
    15
  2. B
    9
  3. C
    135
  4. D
    10
View written solutionFree

Correct answer: D

  1. We are given
P=[100310931]P=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix}P=​139​013​001​​

and matrices P,QP,QP,Q satisfy Q−P5=I3.Q-P^5=I_3.Q−P5=I3​. So, Q=P5+I3.Q=P^5+I_3.Q=P5+I3​.

We need q21+q31q32.\frac{q_{21}+q_{31}}{q_{32}}.q32​q21​+q31​​.

  1. Write PPP as P=I+N,P=I+N,P=I+N, where
N=[000300930].N=\begin{bmatrix}0&0&0\\3&0&0\\9&3&0\end{bmatrix}.N=​039​003​000​​.

Since NNN is strictly lower triangular of order 333, we have N3=0.N^3=0.N3=0. Therefore, P5=(I+N)5=I+5N+10N2.P^5=(I+N)^5=I+5N+10N^2.P5=(I+N)5=I+5N+10N2.

  1. Compute N2N^2N2.
N=[000300930]N=\begin{bmatrix}0&0&0\\3&0&0\\9&3&0\end{bmatrix}N=​039​003​000​​

Then

N2=[000000900].N^2= \begin{bmatrix}0&0&0\\0&0&0\\9&0&0\end{bmatrix}.N2=​009​000​000​​.
  1. Now compute
10N2=[0000009000].\qquad 10N^2=\begin{bmatrix}0&0&0\\0&0&0\\90&0&0\end{bmatrix}.10N2=​0090​000​000​​.

Hence

P5=I+5N+10N2=[1001510135151].P^5=I+5N+10N^2 =\begin{bmatrix}1&0&0\\15&1&0\\135&15&1\end{bmatrix}.P5=I+5N+10N2=​115135​0115​001​​.
  1. Since Q=P5+I3,Q=P^5+I_3,Q=P5+I3​, we get
Q=[2001520135152].Q=\begin{bmatrix}2&0&0\\15&2&0\\135&15&2\end{bmatrix}.Q=​215135​0215​002​​.

So, q21=15,q31=135,q32=15.q_{21}=15,\quad q_{31}=135,\quad q_{32}=15.q21​=15,q31​=135,q32​=15.

  1. Therefore,
q21+q31q32=15+13515=15015=10.\frac{q_{21}+q_{31}}{q_{32}} =\frac{15+135}{15} =\frac{150}{15}=10.q32​q21​+q31​​=1515+135​=15150​=10.
  1. Checking options:
  • A: 151515 ❌
  • B: 999 ❌
  • C: 135135135 ❌
  • D: 101010 ✅

So the correct answer is D.

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