List of top Mathematics Questions on Matrices asked in TS PGECET

Rank of the matrix \[ A= \begin{bmatrix} 1& 0& 1& 1\\ 0& 1&-3&-1\\ 3& 1& 0& 2\\ 1& 1&-2& 0 \end{bmatrix} \] is

Let \(A=[a_{ij}]\) for \(i,j=1,2,3,\ldots,n\) be an \(n\times n\), \((n>1)\) matrix such that \[ a_{ij}= \begin{cases} i!, & \text{if } i=j, j, & \text{if } i>j, 0, & \text{if } i<j. \end{cases} \] Which one of the following is correct?

Consider the system of equations \[ x+y+z=4\mu,\qquad x+2y+2z=10\mu,\qquad x+3y+4\lambda z=\mu^2+15, \] where \(\lambda\) and \(\mu\) are real numbers. Which of the following is correct?

Rank of the matrix \[ A= \begin{bmatrix} 1& 0& 1& 1 0& 1&-3&-1 3& 1& 0& 2 1& 1&-2& 0 \end{bmatrix} \] is

Let \(A=[a_{ij}]\) for \(i,j=1,2,3,\ldots,n\) be an \(n\times n\), \((n>1)\) matrix such that \[ a_{ij}= \begin{cases} i!, & \text{if } i=j, j, & \text{if } i>j, 0, & \text{if } i<j. \end{cases} \] Which one of the following is correct?

Consider the system of equations \[ x+y+z=4\mu,\qquad x+2y+2z=10\mu,\qquad x+3y+4\lambda z=\mu^2+15, \] where \(\lambda\) and \(\mu\) are real numbers. Which of the following is correct?

If $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 2 \\ 0 & 1 & 5 \end{bmatrix}$, then $\left|P^{-1}AP - 2I\right| =$}

The system of equations $2x + y = 2z$, $y + z = 0$, $3y + \alpha z = x$ has:

If \( \lambda_1, \lambda_2 \) and \( \lambda_3 \) are the eigen values of a square matrix \( A \), then the eigen values of \( A^{-1} + 2I + A \) are

Let the matrix \( A = \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix} \).
Statement-I: \( A^2 = 9I \)
Statement-II: The eigen values of \( A \) are \( -3 \) and \( 3 \)
The correct answer is

If $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 2 \\ 0 & 1 & 5 \end{bmatrix}$, then $\left|P^{-1}AP - 2I\right| =$}

The system of equations $2x + y = 2z$, $y + z = 0$, $3y + \alpha z = x$ has:

Let the matrix \( A = \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix} \).
Statement-I: \( A^2 = 9I \)
Statement-II: The eigen values of \( A \) are \( -3 \) and \( 3 \)
The correct answer is

If \( \lambda_1, \lambda_2 \) and \( \lambda_3 \) are the eigen values of a square matrix \( A \), then the eigen values of \( A^{-1} + 2I + A \) are