Matrices practice · Mixed difficulty

Option A (4) is correct because after setting up the matrix A and using the condition A²=I, we find that 3a²+4b² = 4.

Option A ($x=-1, y=0$) is correct because by simplifying the determinant and equating it to $x+iy$, we find $x=-1$ and $y=0$.

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the determinant can be expanded using cofactors of any row or column.

Option B ($\left( {x - p} \right)\left( {x - q} \right)\left( {x + p + q} \right)$) is the correct factorization of the given determinant.

Option D (-1) is correct because expanding the determinant and using the given condition $\Delta = 0$ leads to the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = -1$.

Option A ((-6,11)) is correct because the values of c and d are found by comparing the given equation with the characteristic equation of matrix A.

The correct answer is C (0) because the determinant simplifies to zero after using the substitution $u = \frac{1}{t}$ and finding that $A + B = 0$.

Option C (1) is correct because the determinant's square equals 1, derived from the given orthogonality conditions and normalization of vectors.

Option C (5) is correct because using Pascal's rule and determinant properties, we find that $m=5$ satisfies the condition for the determinant to be zero.

Option D is correct because there exist infinitely many matrices B such that AB = BA when a = b, where a and b are elements of matrix B.

Option D is correct because by calculating A² and substituting into the given equation, we can solve for alpha and beta, and then find alpha + beta.

Option D (0) is correct because the sum of the determinants results in a determinant with two identical rows, making its value zero.

Option D is correct because the system has non-trivial solutions for exactly three values of $\lambda$: 0, 1, and -1.

Option D ($\Delta \in[2,4]$) is correct because the determinant simplifies to 2(sin²$\theta$ + 1), which ranges from 2 to 4.

Option C ($AB = BA$) is correct because the expansion of $(A + B)^2$ results in $A^2 + AB + BA + B^2$, which is only equal to $A^2 + 2AB + B^2$ if $AB = BA$.

The correct answer is D (divisible by both $x$ and $y$). Simplifying the determinant results in $D = xy$, which is clearly divisible by both $x$ and $y$.

Option A (Unique solution satisfying $x+y=1$) is correct because after finding the value of k for infinitely many solutions, the resulting system has a unique solution with x+y=1.

Option B ($2s^3$) is correct because after applying column operations and simplifying, the determinant evaluates to $2s^3$.

Option A ($k \ne 0$) is correct because for a unique solution, the determinant of the coefficient matrix must be non-zero, which implies k is not equal to 0.

Option D is correct. The matrix elements are defined by a_ij = i + j, and substituting the row and column indices gives the correct matrix.

Option C (r ≤ min(m,n)) is correct because the rank of a matrix cannot exceed the minimum of its number of rows and columns.

Option B ($2 n \pi \pm \frac{\pi}{3}, n \in {I}$) is correct because it satisfies the condition $A + A' = I$ when solving for alpha.

The correct answer is A. Given that AB = I, multiplying the matrices and equating the elements gives x + y = 0.

Option B ($A^{-1}=|A|^{-1}$) is not true because $A^{-1}$ is a matrix, while $|A|^{-1}$ is a scalar. The correct formula is $A^{-1} = \frac{1}{|A|} adj(A)$.

Option B (Both Assertion and Reason are true but Reason is not the correct explanation of Assertion) is correct. The assertion is true because if A is symmetric, $X^TAY = Y^TAX$. The reason is also true, but it's not the correct explanation.

Option A ($2,2,3,4$) is correct because solving the system of equations derived from the matrix equality yields these values for $x, y, z,$ and $w$.

Option A (0) is correct because the function f(x) simplifies to 0, making f(50) also equal to 0.

Option D (512) is correct because each of the 9 entries in the 3x3 matrix can be either 2 or 0, leading to $2^9 = 512$ possible matrices.

The correct answer is C (Sphere) because simplifying the determinant leads to the equation $x^2 + y^2 + z^2 = 4$, which represents a sphere.

Option A, $(\mathrm{A}+\mathrm{B})+\mathrm{C}=\mathrm{A}+(\mathrm{B}+\mathrm{C})$, is correct because it represents the associative law for matrix addition.