Test by Bikram | Mock GATE | Test 3 | Question: 15

Question

$’X’$ and $’T’$ are two square matrices. 

$’X’$ has eigen values $3, 0, 2$.   $’T’$ has eigen values $4$ and $1$.

Which of the following statements is CORRECT?

• A.     $X$ and $T$ both are invertible.
• B.     $T$ is invertible but not $X$.
• C.     $X$ is invertible but not $T$.
• D.     None of them are invertible.

Comments

A matrix has an eigenvalue 0 iff it is singular.

X has an eigenvalue 0. => X is singular. => X isn't invertible. (A matrix can either be singular or invertible. Not none, not both)

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T has eigenvalues 4 and 1.

Hence, determinant of T = product of eigenvalues = 4.

Which is non zero.

=> Invertible.

Answer 1

If any of the eigenvalues is 0, that means that the determinant of that matrix is zero. 

If the determinant is 0, then that matrix is singular, and it's inverse doesn't exist.

So, here (B)T is invertible but not X

Answer 2

The matrix is invertible iff it has not 0 as eigen value.