Q.5 (a) For the matrix A=[a_ij]_(3×3) , where a_ij=i+j , find the rank.

A=beginbmatrix2&3&4\\3&4&5\\4&5&6endbmatrix R₂→ R₂-R₁,\ R₃→ R₃-R₁ : beginbmatrix2&3&4\\1&1&1\\2&2&2endbmatrix ; R₃→ R₃-2R₂ : beginbmatrix2&3&4\\1&1&1\\0&0&0endbmatrix R₁rightarrow R₂,\ R₂→ R₂-2R₁ : beginbmatrix1&1&1\\0&1&2\\0&0&0endbmatrix (echelon form, 2 non-zero rows). |A|=0 , but the 2×2 minor beginvmatrix2&3\\3&4endvmatrix=-1≠0 . Rank of A = 2.

For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank.

Q.5 (a) For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank.

Doubtly Asked 25 Oct 2026 mu engineering Save WhatsApp Report

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\(A=\begin{bmatrix}2&3&4\\3&4&5\\4&5&6\end{bmatrix}\)

\(R_2\to R_2-R_1,\ R_3\to R_3-R_1\): \(\begin{bmatrix}2&3&4\\1&1&1\\2&2&2\end{bmatrix}\); \(R_3\to R_3-2R_2\): \(\begin{bmatrix}2&3&4\\1&1&1\\0&0&0\end{bmatrix}\)

\(R_1\leftrightarrow R_2,\ R_2\to R_2-2R_1\): \(\begin{bmatrix}1&1&1\\0&1&2\\0&0&0\end{bmatrix}\) (echelon form, 2 non-zero rows).

\(|A|=0\), but the 2×2 minor \(\begin{vmatrix}2&3\\3&4\end{vmatrix}=-1\ne0\). Rank of A = 2.

Doubtly AI · 25 Oct 2026 AI-verified · Oct 2026 Report

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For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank.

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