Matrices practice · Mixed difficulty

Option D (539) is correct because it is the result of the matrix multiplication A'BA as shown in the step-by-step solution.

Option A ($\left( {\frac{3}{2},3} \right]$) is correct because by finding the determinant of matrix A and using the given range for $\theta$, we can determine the interval in which det(A) lies.

Option D (1: 1: 1) is correct because when λ = 0, the system of equations simplifies, leading to the ratio x:y:z = 1:1:1.

Option B ((Adj. B) (Adj.A)) is correct because Adj(AB) = Adj(B)Adj(A) is a standard property of adjoint matrices.

Option C (3) is correct because the determinant of the matrix is non-zero, indicating that the matrix is full rank (rank 3).

Option C (1) is the correct answer because the determinant equation simplifies to $(\sin x - \cos x)^2(\sin x + 2\cos x) = 0$, and within the given interval, only $\tan x = 1$ is a valid solution, giving one real solution.

Option B (1 x 1) is correct because the resulting matrix will be a 1x1 matrix (a scalar).

Option B (0) is correct because the determinant is zero when two columns are identical after applying column operations.

Option D (2) is correct because after applying row operations and simplifying the determinant equation, the value of the expression $\frac{a}{x} + \frac{b}{y} + \frac{c}{z}$ is found to be 2.

Option D (No real values) is correct because there is no real value of alpha that satisfies the condition A² = B.

Option C (Skew-symmetric matrix) is correct because the transpose of (AB - BA) is equal to -(AB - BA), satisfying the definition of a skew-symmetric matrix.

Option A ($\alpha$) is correct because the determinant simplifies to an expression independent of $\alpha$.

Option A ($\lambda \ne - 17$) is correct because the matrix is invertible if its determinant is not equal to zero, which leads to this condition on lambda.

Option D (0) is correct because after multiplying each row by a, b, and c respectively, and factoring out abc, two columns become identical, making the determinant zero.

Option C (-9,14) is the correct answer because it satisfies the matrix equation $A^2+xA+yI=0$ when substituting the given matrix A.

Option B (1) is correct because the system has no solution when the determinant of the coefficient matrix is zero and the adjoint of the coefficient matrix times the constant matrix is non-zero, which occurs for only one value of k.

Option B ($A^2+B^2$) is correct because given $B=-A^{-1}BA$, it implies $AB = -BA$, and thus $(A+B)^2$ simplifies to $A^2 + B^2$.

The correct answer is A (36) because it is the modulus of the sum of all possible values of the determinant of the symmetric matrix P, derived from the given conditions.

Option B (divisible by both $x$ and $y$) is correct because the determinant simplifies to $xy$, which is divisible by both $x$ and $y$.

The correct answer is A ($\Delta_1+\Delta_2=0$) because $\Delta_2$ can be transformed into $-\Delta_1$ by applying row operations and factoring.

Option D ($3^{\frac{n(n-1)}{2}}$) is correct because each of the $\frac{n(n-1)}{2}$ elements above the diagonal can take 3 values (0, 1, -1), determining the rest of the skew-symmetric matrix.

Option D (${k^n}$) is correct because the determinant of a scalar matrix A with diagonal elements equal to k is k raised to the power of the order of the matrix, which is n.

Option A (0) is correct because the determinant of a skew-symmetric matrix of odd order is always zero.

Option D ($125 \Delta$) is correct because multiplying each element of a third-order determinant by 5 multiplies the determinant's value by $5^3 = 125$.

Option C ($AB = O$ implies $A = O$ or $B = O$) is not true because the product of two non-zero matrices can be a zero matrix.

Option C (An invertible matrix) is correct because the given equation implies that A has an inverse, which is $A^4 - A^2$.

Option C ($-4,1$) is correct because these values satisfy the equation $A^2 + aA + bI = O$.

Option D (is inconsistent when |a| = √3) is correct because when |a| = √3, the determinant of the coefficient matrix is zero, while at least one of the determinants formed by replacing a column with the constants is non-zero, indicating no solution.

Option D (More than 2) is correct because there are infinitely many matrices that satisfy X² = I, beyond just the identity matrix itself.

Option D (None of these) is correct because the calculated matrix $D$ does not match any of the given options.