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GATE 2026 BT – Question 41

Engineering Mathematics · Linear Algebra: Matrices and determinants, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple select

For the given matrix, which of the statements given below is/are true?

$\begin{bmatrix} 3 & 1 & 5 \\ 2 & -1 & 0 \\ 5 & 2 & 9 \end{bmatrix}$

  1. The matrix is full rank
  2. The matrix is invertible
  3. The matrix has a determinant
  4. The matrix has a transpose

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Show answer and explanation

Correct answer: (C) The matrix has a determinant; (D) The matrix has a transpose

Explanation

The determinant is $3(-9 - 0) - 1(18 - 0) + 5(4 + 5) = -27 - 18 + 45 = 0$. The third row is the sum of the first two, so the matrix is singular, with rank 2: it is not full rank and not invertible. A square matrix still has a determinant (equal to 0) and a transpose.