Matrices practice · Medium

Option C (0) is correct because the determinant can be expressed as a product of two determinants, each having a column of zeros, resulting in a zero determinant.

The correct answer is 0.5 because after simplifying the determinant and using the given conditions, we can find the value of $\frac{\sqrt{2} p}{q}+1$.

The correct answer is C (0) because after factoring and applying column operations, two columns become identical, resulting in a zero determinant.

Option B (I) is correct because given the condition M^-1 = adj(adj(M)), it can be shown that M² = I, and therefore adj(M²) = adj(I) = I.

Option A (1) is correct because the given determinant can be factored into the square of a Vandermonde determinant, which simplifies to $K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2$ with K = 1.

Option C ($(I-A)^{-1}$) is correct because multiplying $(I+A+A^2+\cdots+A^{k-1})$ by $(I-A)$ results in $I-A^k$, which simplifies to $I$ since $A^k=0$.

Option B ($\frac{9}{4}$) is correct because the given condition implies that the triangle is equilateral, and for an equilateral triangle, ${\sin ^2}A + {\sin ^2}B + {\sin ^2}C = \frac{9}{4}$.

Option C (231) is correct because it involves matrix exponentiation and pattern recognition.

Option B (2) is correct because the determinant simplifies to 2 using the trigonometric identity for the sum of angles in a triangle.

Option C (1) is correct because there is only one distinct real root within the given interval after simplifying the determinant equation.

Option C (2) is correct because by differentiating the given equation and substituting x=0, we can solve for the value of lambda.

Option A ($nA-(n-1)I$) is correct because by observing the pattern in the first few powers of A, we can deduce the general formula for A^n.

Option A (0) is correct because the area of the triangle is zero, indicating that the points are collinear. This can be shown by simplifying the determinant used to calculate the area.

Option D (0) is correct because applying column operations results in two identical columns, making the determinant zero.

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

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 B (1) is correct because the determinant of the coefficient matrix must be zero for infinitely many solutions, leading to a cubic equation with only one real root.

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 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 $\left(\begin{array}{c}1 \\ -1 \\ -1\end{array}\right)$ 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 B is correct. By finding the inverse of the matrices surrounding P and performing the matrix multiplications, we can solve for P and find that P = 1/19 * [[48, -25], [-70, 42]].

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).

Option B (-2) is correct because after expanding the determinant, the coefficient of x is found to be -2.

Option A is correct. By expanding the magnitudes and using the properties of complex numbers, the expression simplifies to $32(x_1^2 + y_1^2)$.

Option D (${x^3}$) is correct because after applying properties of cube roots of unity and determinant operations, the determinant simplifies to ${x^3}$.

Option B is correct because it is the inverse of the given matrix A, calculated using the adjoint and determinant of A.

Option A (1) is correct because by expanding $(B+C)^{N+1}$ and using the given conditions, we find that $K=N$, thus $K/N = 1$.

The correct answer is 4 because, after simplification using the given condition, the determinant evaluates to 4a²b²c².

Option A ($I+15 A$) is correct because expanding $(I+A)^4$ using the binomial theorem and the property $A^2=A$ simplifies to $I+15A$.