List of top Mathematics Questions on Matrices

Diagonal elements of a skew-Hermitian matrix are:

If $A$ is a null matrix then

Let $A$ be a symmetric matrix, then

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

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then compute $A^2 - 4A - 5I$.

Find the cost (per kg) of each fertilizer A, B and C that a farmer needs to buy, such that 1 kg each of fertilizer A and C added to 2 kg of B costs him ₹ 400. Also, the cost of each kg of fertilizer B and C added together is equal to the cost of 1 kg of fertilizer A. However, the cost of 3 kg of fertilizer B added to ₹ 200 is the same as the cost of 1 kg of fertilizer A and C together. Use the matrix method to find the solution.

If \( A \) is a non-singular matrix, then which of the following is not true ?

Let \( A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -3 & 0 & 1\\ 2 & 4 & 0 \end{bmatrix} \). If \( A + B + C = O \), then matrix \( C \) is :

If \( A \) is a symmetric matrix, then for any matrix \( B \) of order same as \( A \), \( BAB' \) is a/an :

If \( A = \begin{bmatrix} \sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{bmatrix} \) and \( A + A' = I \), then \( \theta \) is equal to :

If \[ P= \begin{bmatrix} 0 & 1 & 2\\ 2 & 3 & 4\\ 1 & -1 & 0 \end{bmatrix} \] and \[ Q= \begin{bmatrix} 2 & -1 & 5\\ -4 & 2 & -4\\ 2 & 2 & -4 \end{bmatrix}, \] find the matrix product \[ QP. \] Hence, solve the following system of linear equations using matrices: \[ x-y=3,\qquad 2x+3y+4z=17,\qquad y+2z=7. \]

Given that $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, $Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}$ and $R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}$, find a matrix $S$ such that $PQ - RS$ is a null matrix.