Matrices practice · Medium

Option C, ${k^{n - 1}}\;\mathrm{adj\;}A$, is correct because it represents the correct relationship between adj(kA) and adj(A) for a square matrix of order n.

None of these is the correct answer. The correct answer is $\left[\begin{array}{ll}1 & 0 \\25 & 1\end{array}\right]$, which is obtained by induction.

Option B (-8) is correct because the determinant simplifies to -8 after applying column operations.

Option C ($2^{8}$) is correct because we simplify the determinant of the matrix A, then use the properties of adjoints and determinants to find the final value.

Option D ($\beta=-8$) is correct because by repeatedly substituting $A^2 = 3A + \alpha I$ into higher powers of A, we can express $A^4$ in terms of A and I, and then compare coefficients.

Option B (${2^n} - 1$) is correct because by expanding $(I+A)^n$ using the binomial theorem and the property $A^2 = A$, we find that the coefficient of A is ${2^n} - 1$.

Option D (${e^{{\Delta _1}/{\Delta _3}}}$ and ${e^{{\Delta _2}/{\Delta _3}}}$) is correct because using Cramer's rule after taking logarithms yields these expressions for x and y.

Option A (1) is correct because, given B² = 0, the expression simplifies to the determinant of the identity matrix.

Option D is correct. The system has a unique solution when $\alpha=2$, so it cannot have infinitely many solutions.

Option A (20) is correct because by setting the determinant of the coefficient matrix to zero, we find the possible values of theta and then calculate the required sum.

The correct answer is exactly one negative value of $\lambda$ because the system is consistent only when $\lambda = -2/3$, which is a single negative value.

Option A (O) is correct because the expression simplifies to the zero matrix after algebraic manipulation.

Option D (0) is correct because x = 0 satisfies the given equation.

Option B (I, II, III only) is correct because the given conditions and matrix properties lead to the conclusion that (I-A) and (I-B) are singular, and A+B=AB+I.

Option B (infinite number of solutions) is correct because the system of homogeneous equations always has at least the trivial solution (x=y=z=0), and if the determinant of the coefficient matrix is zero, there are infinitely many solutions.

Option C (Has infinitely many solutions for $\alpha=-1$) is correct because when $\alpha=-1$, the determinant of the coefficient matrix is zero, and the system has infinitely many solutions if the equations are consistent.

Option A ($\det A$) is correct because the determinant of $A^{-1} \text{adj} A$ simplifies to $\det A$ using determinant properties.

Option A is correct. AA^T results in the identity matrix I because the given matrix is an orthogonal matrix.

Option C ((-3,-1)) is correct because after simplifying the determinant using column and row operations, the expression becomes $-(2 + \sin 2x)$, and the range of this expression is [-3, -1].

Option D (None of these) is correct because the value of t depends on a, b, and c, and it is equal to $4a^2b^2c^2$.

Option D (24) is correct because by substituting x=1 into the equation and simplifying the determinant, we can solve for the value of A.

Option A ($\left[\begin{array}{cc}1 & 2005 \\ 0 & 1\end{array}\right]$) is correct because the given matrix operations simplify to $A^{2005}$, which can be easily computed.

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

Option A (0) is correct because after applying row operations, two rows become identical, making the determinant zero.

Option B ($\frac{{72}}{{5}},-\frac{{21}}{{5}}$) is correct because for infinitely many solutions, the determinant of the coefficient matrix and the determinant with the z-column replaced must both be zero.

Option A (Reflexive but not symmetric) is correct because the relation contains all (a,a) pairs, but lacks symmetry as (1,2) is present but (2,1) is not.

The correct answer is 0 because the determinant of (P²+Q²) is 0, derived from the given conditions P³=Q³ and P²Q=Q²P.

Option C ($(2+\sqrt{2}, 2-\sqrt{2})$) is correct because it represents the maximum and minimum values of the function f(θ) after simplification and applying trigonometric identities.

Option C (2) is correct. The determinant equation simplifies to $\sin x = 0$ or $\sin x = \cos x$, leading to two distinct real roots within the given interval.

Option C is correct. Using the property that |adj(adj(A))| = |A|^(n-1)² and |adj(adj(adj(A)))| = |A|^(n-1)⁴, we can deduce that k=3.