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CSIR NET Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free to read right now. Pick your subject below.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

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Tagged “linear algebra”

“Same characteristic polynomial ⇒ similar”

False — 0 matrix and [[0,1],[0,0]]

“Diagonalisable ⇒ invertible”
“Real matrix with real eigenvalues is diagonalisable”

False — [[1,1],[0,1]]

“AB = I ⇒ BA = I for all matrices”
“Every real symmetric matrix is positive definite if det > 0”
“Same characteristic and minimal polynomial ⇒ similar”

False — 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

“AB and BA have the same minimal polynomial”

False — A = [[0,1],[0,0]], B = [[0,0],[0,1]]

“Commuting matrices are simultaneously diagonalisable”

False — A = B = [[0,1],[0,0]]

“Real matrix diagonalisable over C⇒\mathbb{C} \Rightarrow diagonalisable over R\mathbb{R}”
“An injective linear operator on a vector space is surjective”

False — The right shift on ℓ2:(x1,x2,… )↦(0,x1,x2,… )\ell^{2}: (x_{1}, x_{2}, \dots) \mapsto (0, x_{1}, x_{2}, \dots)

“Every real matrix has a Jordan form over R\mathbb{R}”

False — The rotation [[0,−1],[1,0]]

“Two matrices with the same characteristic and minimal polynomials are similar”
“A real matrix with all real eigenvalues is orthogonally diagonalisable”

False — [[1,1],[0,1]]

“det A > 0 implies A is positive definite”

False — A = diag(−1, −1)

“There exist matrices with AB − BA = I”

False — Impossible over R\mathbb{R} or C\mathbb{C} in finite dimensions

“Equivalent matrices are similar”

False — A=I2A = I_{2} and B=2I2B = 2I_{2}