Linear Algebra

Suppose that P is a 4x5 matrix such that every solution of the equation Px=0 is a scalar multiple of \begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T. The rank of P is _______

Consider the following matrix.
\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}
The largest eigenvalue of the above matrix is __________.

If x+2 y=30,then \left(\frac{2 y}{5}+\frac{x}{3}\right)+\left(\frac{x}{5}+\frac{2 y}{3}\right) will be equal to

Let A and B be two nxn matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB)=rank (A)rank (B)
II. det(AB)=det(A)det(B)
III. rank(A+B) \leq rank (A) + rank (B)
IV. det(A+B) \leq det(A) + det(B)

Which of the above statements are TRUE?

Consider the following matrix:
\begin{bmatrix} 1 & 2 & 4 & 8\\ 1& 3 & 9 &27 \\ 1 & 4 & 16 &64 \\ 1 & 5 & 25 &125 \end{bmatrix}
The absolute value of the product of Eigenvalues of R is _________ .

Let X be a square matrix. Consider the following two statements on X.

I. X is invertible
II. Determinant of X is non-zero

Which one of the following is TRUE?

Consider a matrix P whose only eigenvectors are the multiples of \begin{bmatrix} 1\\ 4 \end{bmatrix}.
Consider the following statements.
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized
Which one of the following options is correct?

Consider a matrix A=uv^{T}\; where \; u=\begin{bmatrix} 1\\ 2 \end{bmatrix},v=\begin{bmatrix} 1\\ 1 \end{bmatrix} Note that v^{T} denotes the transpose of v. The largest eigenvalue of A is _____.

If A is a skew symmetric matrix then A^{t} is

If the characteristics polynomial of 3x3 matrix M over R ( the set of real numbers) is \lambda ^{3}-4\lambda ^{2}+a\lambda +30,a\in R, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is ________.