Matrices practice · Medium

Option A (0) is correct because applying row operations results in a row of zeros, making the determinant zero.

Option D (x = 0, y = 0) is correct because when the determinant is calculated and expressed in the form x + iy, both x and y are found to be 0.

Option D (${y^3}$) is correct because the determinant simplifies to this expression using properties of determinants and the roots of the given equation.

Option D (-pqr) is correct because the determinant simplifies to -pqr times another determinant.

Option B (64) is correct because it is the value of the derivative of the determinant function evaluated at x=0, calculated using the product rule for determinants.

Option A ($I \cos \theta+J \sin \theta$) is correct because it accurately represents the given matrix B in terms of I and J.

Option D (No real values) is correct because the conditions derived from ${A^2} = B$ lead to contradictory values for $\alpha$, indicating no real solution exists.

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 B (-1) is correct because the given condition implies that abc = -1 after simplifying the determinant equation.

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 ($(I-A) B$) is correct because, after substituting $\tan(\theta / 2) = a$ and performing matrix multiplication, it simplifies to $I+A$.

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 (Statement-1 is true, Statement-2 is false) is correct because Statement-1 is true based on determinant calculation, while Statement-2 is false.

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

Option A (Zero) is correct because when x=0, the determinant evaluates to zero.

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 D (3) is correct because if lambda equals 3, the determinant of the coefficient matrix becomes zero, resulting in no unique solution.

Option D (-8) is correct because after substituting the values of [e], [$\pi$], and [$\pi^2-6$], the determinant of the matrix is -8.

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

Option A ($n$) is correct because the determinant simplifies to an expression that does not contain $n$, indicating it's independent of $n$.

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 (Orthogonal) is correct because the matrix A satisfies the condition A * A_transpose = I, which is the definition of an orthogonal matrix.

Option D (zero) is correct because the determinant evaluates to zero when the terms are in a geometric progression.

Option B ((x - p)(x - q)(x + p + q)) is correct because by applying row operations and factoring, the determinant simplifies to this expression.