List of top Mathematics Questions on Matrix

Three students run on a racing track such that their speeds add up to 6 km/h. However, double the speed of the third runner added to the speed of the first results in 7 km/h. If thrice the speed of the first runner is added to the original speeds of the other two, the result is 12 km/h. Using the matrix method, find the original speed of each runner.

Find the value of $x$, if \[ \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ x \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

If A and B are two square matrices of the same order, then (A + B)(A - B) is equal to:

If the matrix \[ \begin{bmatrix} 1 & 12 & 4y \\ 6x & 5 & 2x \\ 8x & 4 & 6 \end{bmatrix} \]
is a symmetric matrix, then the value of $ 2x + y $ is:

If a matrix A is both symmetric and skew-symmetric, then A is:

Let $ p_A(x) $ denote the characteristic polynomial of a square matrix $ A $. Then, for which of the following invertible matrices $ M $, the polynomial $ p_M(x) - p_{M^{-1}}(x) $ is constant?

Then, which one of the following is TRUE?

Consider the balanced transportation problem with three sources $ S_1, S_2, S_3 $, and four destinations $ D_1, D_2, D_3, D_4 $, for minimizing the total transportation cost whose cost matrix is as follows:

where $ \alpha, \lambda>0 $. If the associated cost to the starting basic feasible solution obtained by using the North-West corner rule is 290, then which of the following is/are correct?

Let $ A = \begin{bmatrix} 2 & 2 + p & 2 + p + q \\4 & 6 + 2p & 8 + 3p + 2q \\6 & 12 + 3p & 20 + 6p + 3q \end{bmatrix} $ If $ \text{det}(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n, \, m, n \in \mathbb{N}, $ then $ m + n $ is equal to:

Let $ A $ be a square matrix of order 3 such that $ \text{det}(A) = -2 $ and $ \text{det}(3 \cdot \text{adj}(-6 \cdot \text{adj}(3A))) = 2^{m+n} \cdot 3^{mn} $, where $ m > n $. Then $ 4m + 2n $ is equal to:

Let $ A $ be a $ 3 \times 3 $ matrix such that $ X^T AX = 0 $ for all nonzero $ 3 \times 1 $ matrices $ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $. \[ A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] If $ \det(\text{adj}(2(A + I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} $, $ \alpha, \beta, \gamma \in \mathbb{N} $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is:

If $ A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} $, then find the value of $ K $ if $ A^2 = 6A + K I_2 $, where $ I_2 $ is the identity matrix.

If $ P $ is a non-singular matrix of order $ 5 \times 5 $ and the sum of the elements of each row is 1, then the sum of the elements of each row in $ P^{-1} $ is:

Let $ f(\theta) = \begin{vmatrix} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{vmatrix} $. Suppose $ A $ and $ B $ are respectively the maximum and minimum values of $ f(\theta) $. Then $ (A, B) $ is equal to:

If $ |a| = 2 $ and $ -3 \leq k \leq 2 $, then $ |a| |k| \in: $

If $ A $ and $ B $ are two non-zero square matrices of the same order such that: $$ (A + B)^2 = A^2 + B^2, $$ then:

The determinant $\begin{vmatrix} x+1 & x-1 \\ x^2+x+1 & x^2-x+1 \end{vmatrix}$ is equal to:

If a matrix has 36 elements, the number of possible orders it can have is:

If $ A $ and $ B $ are two non-zero square matrices of the same order such that \[ (A + B)^2 = A^2 + B^2, \] then:

If $ A = [a_{ij}] $ is an identity matrix, then which of the following is true?

The value of \[ \begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix} \textbf{is:} \]