Matrices practice · Mixed difficulty

Option A (0) is correct because substituting the given condition into the determinant expression results in a value of 0.

Option B (Is a zero matrix) is correct because for a singular matrix A, its determinant is 0, and A adj(A) equals |A|I, which results in a zero matrix.

Option D (2⁸) is correct because after simplifying the matrix A and calculating its determinant, the determinant of adj(adj A²) is found to be 2⁸.

Option A ($(2 n+1) \frac{\pi}{2}$) is correct because the product of the matrices is a null matrix only when $\cos(\theta - \phi) = 0$, which occurs when $\theta - \phi$ is an odd multiple of $\frac{\pi}{2}$.

Option D, $\left[\begin{array}{rr}1 & 0 \\ -n & 1\end{array}\right]$, is correct because after repeated application of the given transformations, $A^T C^n A$ simplifies to $B^n$, which equals the given matrix.

Option D (2I+C) is correct because by manipulating the given equation and using the inverse, we can express C^-2 as 2I+C.

Option A (Symmetric) is correct because the transpose of M'AM is equal to itself, indicating symmetry.

Option B (Symmetric matrix) is correct because the transpose of $A^4$ is equal to $A^4$, which is the definition of a symmetric matrix.

Option A is correct. The expression simplifies to AB, and the matrix multiplication results in the given matrix.

The correct answer is D (None of these) because the determinant can be expressed as a product of two matrices, resulting in a zero determinant.

Option A ($x = - 2,\;y = 1$) is correct because these values satisfy the given matrix equation.

Option D (None of these) is correct because if A has an inverse and satisfies the given relation, then all elements of A must be zero, which contradicts the existence of an inverse. Therefore, A^-1 does not exist.

Option C ($\log _{e} 2$) is correct because the limit can be expressed as a definite integral which evaluates to $\log_e 2$.

Option B (4) is correct because the sum of the series can be found by recognizing the pattern in the trace of the matrix and summing the resulting geometric series.

Option A ($2ac$) is correct because for a non-trivial solution, the determinant of the coefficient matrix must be zero, which simplifies to ab + bc = 2ac.

Option B (-1) is correct because for a unique solution, the determinant of the coefficients of the first two equations must be non-zero, and the third equation must not be a linear combination of the first two. Solving the system leads to k = -1.

Option D (at least one of det(A²⁺B²) or det(A-B) must be zero) is correct because the given conditions lead to (A²⁺B²)(A-B) = 0, implying at least one determinant must be zero.

Option B ($3abc - a^3 - b^3 - c^3$) is correct because the determinant simplifies to this expression after applying column and row operations.

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because matrix addition is a binary operation and is commutative, but commutativity doesn't explain why it's a binary operation.

Option B (0) is correct because the determinant has a column of zeros after applying the column operation C1 -> C1 + C2 + C3, using the property 1 + w + w² = 0.

Option A (Null matrix) is correct because when $\alpha$ and $\beta$ differ by an odd multiple of $\pi/2$, the product E($\alpha$)E($\beta$) results in a null matrix.

Option A (Skew-symmetric matrix) is correct because if A is a skew-symmetric matrix of order 3, then A' = -A, and (A³)' = (A')³ = (-A)³ = -A³, which means A³ is also skew-symmetric.

Option D $\left(\begin{array}{c}1 \\ -1 \\ -1\end{array}\right)$ is correct because it is the result of calculating $A^{-1}\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)$ where A is the given matrix.

Option D (None of these) is correct because the result depends on whether n is even or odd; A^n will be symmetric if n is even and skew-symmetric if n is odd, so it's not consistently one or the other.

Option B (1) is correct because the matrix is not invertible if its determinant is zero, and solving for 'a' when the determinant is zero gives a = 1.

Option D (A^n = nA - (n-1)I) is correct because it can be proven by mathematical induction that this formula holds for all n >= 1, given the matrix A.

Option A is correct because the adjugate of a 2x2 matrix is found by swapping the diagonal elements and changing the sign of the off-diagonal elements.

Option C is correct. Given AB = A and BA = B, multiplying A by A results in A, and multiplying B by B results in B, indicating they are idempotent matrices.

Option B ($ax(2a + 3x)$) is correct because after expanding the determinant and simplifying $f(2x) - f(x)$, the result matches this expression.

Option D is correct. The determinant of a skew-symmetric matrix of odd order is zero, and the determinant of the transpose of a matrix is equal to the determinant of the matrix, but det(-A) = (-1)^n det(A).