Matrices practice · Mixed difficulty

Option D (0) is correct because row operations result in two proportional rows, making the determinant zero.

Option B ($e^2$) is correct because by calculating the determinant of A and solving the resulting equation, we find $x = e^2$.

Option C is correct. For a skew-symmetric matrix A, A^T = -A. Therefore, the expression simplifies to zero.

Option A (1a d-b c[\beginarrayrrd & -b -c & a\endarray]$\frac{{1}}{{a d-b c}}\left[\begin{{array}}{{rr}}d & -b \\ -c & a\end{{array}}\right]$) is correct because the inverse of a 2x2 matrix is given by 1/determinant times the adjugate of the matrix.

Option B ($B^2=I$) is correct because given $A^2 = A$ and $B = I - A$, it follows that $B^2 = (I-A)(I-A) = I - 2A + A^2 = I - 2A + A = I - A = B$, not $I$.

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 (${\log _3}5.{\log _5}81$) is correct because after simplifying both determinants and multiplying them, the result is equal to ${\log _3}5.{\log _5}81 = 4$.

The correct answer is -2 because when $\lambda = -2$, the system of equations becomes inconsistent, leading to no solution.

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 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 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 C ((2,3,4)) is correct because these values satisfy the given system of equations when the matrices are equated.

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

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 C (0) is correct because the determinant simplifies to zero after applying properties of determinants and recognizing that two rows become identical.

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 D (None of these) is correct because the determinant simplifies to 0 since alpha + beta + gamma = 0.

Option C ($n+1$) is correct because by simplifying the determinant and comparing terms, we find that $a = n+1$.

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