TS EAMCET · Maths · Matrices

Suppose $n > 1$ and $A$ is a non-singular matrix of order $n$ such that $|a d j A| = |a d j (a d j A)|$ . Then the matrix whose rank is $n$ , is

A $[\begin{array}{l} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 6 & 7 & 8 \end{array}]$
B $[\begin{array}{l} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{array}]$
C $[\begin{matrix} 1 & 2 & 0 & - 1 \\ 3 & 4 & 1 & 2 \\ - 2 & 3 & 2 & 5 \end{matrix}]$
D $[\begin{matrix} 1 & 4 & - 1 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{matrix}]$

Verified Solution

Answer & Solution

Correct Answer

(A) $[\begin{array}{l} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 6 & 7 & 8 \end{array}]$

Step-by-step Solution

Detailed explanation

We know that adjA=An-1 & adjadjA=An-12, where n is the order of the matrix A. Given adjA=adjadjA ⇒An-1=An-12 ⇒n-1=n-12 ⇒n-1n-1-1=0 ⇒n-1n-2=0 ⇒n=1 (not possible as n>1) or n=2 So, n=2. Rank of the matrix refers to the highest number…

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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Suppose n>1 and A is a non-singular matrix of order n such that adjA=adjadjA. Then the matrix whose rank is n, is

Step-by-step Solution

Suppose $n > 1$ and $A$ is a non-singular matrix of order $n$ such that $|a d j A| = |a d j (a d j A)|$ . Then the matrix whose rank is $n$ , is