GATE CE 2023 Set 2 — Question 25
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Engineering Mathematics → Linear Algebra → Consistency of Linear Systems
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Question
For the matrixwhich of the following statements is/are TRUE?
Correct answer
(B) [A]\x\ = \b\ does not have a unique solution; (C) [A] has three linearly independent eigenvectors
Solution
1.Determinant Check: Calculate the determinant of matrix :
Since , the matrix is singular. A unique solution for exists only if . Therefore, statement (A) is false and statement (B) is true.2.Eigenvectors: Matrix is a real symmetric matrix (). Real symmetric matrices are always diagonalizable, which means they possess linearly independent eigenvectors (where is the dimension of the matrix). Thus, statement (C) is true.
3.Positive Definiteness: For a matrix to be positive definite, all its eigenvalues must be strictly greater than zero. Since , at least one eigenvalue is zero. Therefore, the matrix is positive semi-definite, not positive definite. Statement (D) is false.
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