List of top Engineering Mathematics Questions on Algebra

P and Q are square matrices. Consider the following:
X: $(P^{-1})^{-1} = P$
Y: Symmetric if $Q = -Q^T$
The correct choice is:
Given are two infinite series
P: $\displaystyle \sum \frac{n^2 + 1}{n^2}$
Q: $\displaystyle \sum \left(1 + \frac{1}{n}\right)^{-n}$
The correct choice is:
The coefficient of \(x^4\) in the polynomial \((x-1)^3(x-2)^3\) is equal to __________.
Let \[ M = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix} \] Which of the following are TRUE?
Let \( A \) be a real non-zero square matrix of order \( n \). If the homogeneous system of linear equations \( Ax = 0 \) has only the trivial solution, then:
Let \[ A = \begin{pmatrix} 2 & 0 & 1 \\ 1 & 2 & 5 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix} \] . Then the sum of the geometric multiplicities of the distinct eigenvalues of \( A \) is equal to ________.

Let \( A \) and \( B \) be \( n \times n \) matrices with real entries. Consider the following statements:

  • P: If \( A \) is symmetric, then rank(\( A \)) = Number of nonzero eigenvalues (counting multiplicity) of \( A \)
  • Q: If \( AB = 0 \), then rank(\( A \)) + rank(\( B \)) \( \leq n \)

Then: