For the matrix [A]=bmatrix1&-1&0\\-1&2&-1\\0&-1&1bmatrix, which of the following statements is/are TRUE?

$[A]\{x\}=\{b\}$ has a unique solution

$[A]$ is a positive definite matrix

GATE 2023 CE (CE2) – Question 25

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple select

For the matrix $[A]=\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}$, which of the following statements is/are TRUE?

$[A]\{x\}=\{b\}$ has a unique solution
$[A]\{x\}=\{b\}$ does not have a unique solution
$[A]$ has three linearly independent eigenvectors
$[A]$ is a positive definite matrix

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Correct answer: (B) $[A]\{x\}=\{b\}$ does not have a unique solution; (C) $[A]$ has three linearly independent eigenvectors

Explanation

**Eigenvalues.** For the matrix $A=\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}$ the characteristic polynomial is
$$\lambda(\lambda-1)(\lambda-3)=0 ,$$
so the eigenvalues are $0,\ 1,\ 3$.

**A, unique solution of $A\mathbf x=\mathbf b$:** an eigenvalue of 0 means $\det A=0$, so $A$ is singular and the solution cannot be unique (it is either absent or non-unique). False.
**B, does not have a unique solution:** true, for the same reason. ✓
**C, three linearly independent eigenvectors:** $A$ is real and symmetric, so it has an orthogonal set of three eigenvectors. True. ✓
**D, positive definite:** it would need all eigenvalues $>0$, but one is 0 (the matrix is only positive semi-definite). False.

Answer **B and C**.