Matrices practice · Easy

Option A ($y=2 x$) is correct because solving the matrix equation AB = [6, 8] results in the relationship y = 2x.

Option B (1) is correct. The determinant of the inverse of a matrix is the reciprocal of the determinant, and the determinant of the adjoint is the determinant raised to the power of (n-1).

Option D is correct. X, the symmetric matrix, is calculated as (A + A^T)/2, which results in the given matrix.

The correct answer is C (-6) because if the rank of the matrix is 1, then all 2x2 determinants formed from the matrix must be zero.

Option B (1) is correct because the value of $\lambda$ is 1 when the determinant of the coefficient matrix is zero and at least one of $\Delta x, \Delta y, \Delta z$ is non-zero, indicating no solution.

Option C is correct. By calculating A² and comparing it to B, we find that there is no real value of alpha that satisfies the condition.

Option B (Skew-symmetric matrix of odd order is singular) is correct because the determinant of a skew-symmetric matrix of odd order is always zero, making it singular.

Option A (1) is correct because for A to be skew-symmetric, all diagonal elements must be zero, and this condition is satisfied only when λ = 1.

Option D $\lambda \neq 0$ is correct because for $A^{-1}$ to exist, the constant term $\lambda$ in the polynomial equation must be non-zero.

Option B (10) is correct because by applying logarithm properties and simplifying the expression, the value of the determinant multiplication is 10.

Option A ($f(50)=1$) is correct because after applying row operations, the determinant simplifies to a constant value of 1, independent of x.

Option A (AP) is correct because after simplifying the determinant, we find that 2b = a + c, which is the condition for a, b, c to be in Arithmetic Progression.

Option A (1/2) is correct because the maximum value of the determinant is 1/2, which occurs when sin(2$\theta$) = 1.

Option D ($-2(a+b+c)$) is correct because after applying row and column operations, the determinant simplifies to $(a+b+c)^3$, leading to $x = -2(a+b+c)$.

Option C (2) is correct because solving the system of equations and using the collinearity condition leads to two possible values for $\alpha$, and the sum of their absolute values is 2.

Option C ($adj(A)$) is correct because the inverse of matrix A is equal to its adjoint since the determinant of A is 1.

Option D ((A+B)⁻¹ = B⁻¹+A⁻¹) is correct because the inverse of a sum of matrices is generally NOT equal to the sum of their inverses.

Option A (A null matrix) is correct because as n approaches infinity, the terms cos(nθ)/n and sin(nθ)/n approach 0.

Option C $\left[ {\begin{array}{cccccccccccccccccccc} { - 3}&0\\ 0&{ - 3} \end{array}} \right]$ is correct because solving the system of matrix equations yields this result for matrix X.

Option B is correct because for A+B to be defined, A and B must have the same order, and for AB to be defined, the number of columns of A must equal the number of rows of B. This is only possible if A and B are square matrices of the same order.

Option B is correct because the matrix multiplication of A with itself results in the given expressions for alpha and beta.

Option A ($2g + h + k = 0$) is correct because it represents the condition for the system of linear equations to be consistent, derived by setting the determinant of the augmented matrix to zero.

Option D (∑i a_ii) is correct because the trace of a matrix is defined as the sum of its diagonal elements, which are represented as a_ii.

Option A (2) is correct because A squared is the null matrix, indicating a nilpotent matrix of index 2.

Option B ($A-A^{\top}$) is correct because the transpose of a skew-symmetric matrix is its negative.

Option A ($A$ is symmetric) is correct because $a_{ij} = \cos(i+j) = \cos(j+i) = a_{ji}$, which satisfies the condition for a symmetric matrix.

Option A (0) is correct because the determinant is zero due to the sum of the cube roots of 8 being zero.

Option B (Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1) is correct because Tr(A) = 0 and det(A) = -1, but the determinant doesn't directly explain the trace.

Option A ([\beginlll3 & 1 & 2 3 & 0 & 2 1 & 0 & 1\endlll]$\left[\begin{{lll}}3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1\end{{lll}}\right]$) is correct. By performing column operations on the matrix equation to transform the matrix multiplied by A into the identity matrix, we can directly find $A^{-1}$.

Option C ($\mathrm{A}^2=\mathrm{A}$ and $\mathrm{B}^2-\mathrm{B}$) is correct because given the conditions AB=A and BA=B, we can derive that A and B are idempotent matrices.