Let P1=I=100010001, P2=100001010, P3=010100001, P4=010001100, P5=001100010,P6=001010100, and X=∑k=16PK213102321PKT where PKT denotes the transpose of the matrix PK. Then which of the following options is/are correct?

(A) X-30I is an invertible matrix

JEE Advanced · Mathematics · 18. Matrices

Let $P_{1} = I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], P_{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}], P_{3} = [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}],$ $P_{4} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}], P_{5} = [\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], P_{6} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}],$ and
$X = \sum_{k = 1}^{6} P_{K} [\begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{matrix}] P_{K}^{T}$
where $P_{K}^{T}$ denotes the transpose of the matrix $P_{K} .$ Then which of the following options is/are correct?

A $X - 30 I$ is an invertible matrix
B The sum of diagonal entries of $X$ is $18$
C If $X [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = α [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}],$ then $α = 30$
D $X$ is a symmetric matrix

Verified Solution

Answer & Solution

Correct Answer

(B) The sum of diagonal entries of $X$ is $18$

Step-by-step Solution

Detailed explanation

∵ \begin{matrix} P_{1} = P_{1}^{T} = P_{1}^{- 1}, & P_{3} = P_{3}^{T} = P_{3}^{- 1}, & P_{5} = P_{5}^{T} = P_{5}^{T}, \\ P_{2} = P_{2}^{T} = P_{2}^{- 1}, & P_{4} = P_{4}^{T} = P_{4}^{T}, & P_{6} = P_{6}^{T} = P_{6}^{- 1} \end{matrix}

Now
Let

Q = [\begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{matrix}]

X = \sum_{K = 1}^{6} P_{K} Q P_{K}^{T}

\Rightarrow \begin{matrix} T r a c e (X) = \sum_{K = 1}^{6} T_{r} ((P_{K} Q) P_{K}^{T}) & (∵ T_{r} (A B) = T_{r} (B A)) \end{matrix}

\begin{matrix} = \sum_{K = 1}^{6} T_{r} (P_{K}^{T} P_{K} Q) & (∵ P_{K}^{T} P_{K} = I) \end{matrix}

= \sum_{K = 1}^{6} T_{r} (Q)

= 6 T_{r a c e} (Q)

= 6 \times 3

= 18

\Rightarrow X = \sum_{K = 1}^{6} (P_{K} Q P_{K}^{T})

X^{T} = \sum_{K = 1}^{6} {(P_{K} Q P_{K}^{T})}^{T}

\begin{matrix} X^{T} = \sum_{K = 1}^{6} {(P_{K}^{T})}^{T} Q^{T} {(P_{K})}^{T} & (∵ {(A^{T})}^{T} = A) \end{matrix}

\begin{matrix} X^{T} = \sum_{K = 1}^{6} P_{K} Q P_{K}^{T} & (∵ Q^{T} = Q) \end{matrix}

X^{T} = X

\Rightarrow

Let

R = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]

then

X R = \sum_{K = 1}^{6} P_{K} Q P_{K}^{T} R

\begin{matrix} X R = \sum_{K = 1}^{6} P_{K} Q P_{K} R & (∵ P_{K}^{T} = P_{K}) \end{matrix}

\begin{matrix} X R = \sum_{K = 1}^{6} P_{K} Q R & (∵ P_{K} R = R) \end{matrix}

\begin{matrix} X R = \sum_{K = 1}^{6} P_{K} [\begin{matrix} 6 \\ 3 \\ 6 \end{matrix}] & (∵ Q R = [\begin{matrix} 6 \\ 3 \\ 6 \end{matrix}]) \end{matrix}

\begin{matrix} X R = [\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}] [\begin{matrix} 6 \\ 3 \\ 6 \end{matrix}] & (∵ \sum_{K = 1}^{6} P_{K} = [\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}]) \end{matrix}

X R = [\begin{matrix} 30 \\ 30 \\ 30 \end{matrix}] \Rightarrow X R = 30 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] \Rightarrow α = 30

\Rightarrow X R = 30 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]

X R = 30 R

(X - 30 I) R = 0

Now

|x - 30 I|

must be zero else

R = 0

|x - 30 I| \neq 0

then

{(x - 30 I)}^{- 1} (x - 30 I) R = 0

R = 0

not possible
Hence,

|x - 30 I| = 0

X - 30 I

is not invertible.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let P1=I=100010001, P2=100001010, P3=010100001, P4=010001100, P5=001100010,P6=001010100, andX=∑k=16PK213102321PKTwhere PKT denotes the transpose of the matrix PK. Then which of the following options is/are correct?

Step-by-step Solution