Matrices practice · Easy

The correct answer is A (4) because there are only 4 symmetric matrices with integer entries that satisfy the given condition.

Option A (0) is correct because the determinant is 0 due to proportional rows after simplification.

Option C $\left[ {\begin{array}{cccccccccccccccccccc} {{{a^n}}}&0&0\\ 0&{{{a^n}}}&0\\ 0&0&{{{a^n}}} \end{array}} \right]$ is correct because raising a diagonal matrix to the power of n simply raises each diagonal element to the power of n.

Option C (ax(2a+3x)) is correct because after applying row operations, the determinant simplifies to a(a+x)², and then f(2x) - f(x) = ax(2a+3x).

Option C ($2 \lambda + \mu = 14$) is correct because substituting the values of lambda and mu that result in infinitely many solutions into the equation yields a true statement.

Option D (0) is correct because the determinant of the matrix ${A^3} + {B^3}$ is equal to 0, which is derived by manipulating the given matrix equations.

Option C is correct because it results from correctly isolating X and performing the matrix operations.

Option D (None of these) is correct because the inverse matrix calculation results in a different expression than the given options.

Option B (3) is correct. The condition for non-trivial solution implies that the determinant of the coefficient matrix is zero, leading to the relation $\alpha = \beta = \gamma$, and thus $\frac{a^2}{b} = 3$.

Option D (2) is correct because expanding the determinant and solving for x yields x = 2.

Option D (${\left( {a + b + c} \right)^3}$) is correct because the determinant simplifies to this expression after row and column operations.

Option D ($x=0, y=0$) is correct because the determinant simplifies to zero, meaning x and y are both zero.

Option D ($(b / a)$ is one of cube root of -1) is correct because the determinant equation simplifies to $a^3 + b^3 = 0$, leading to $(b/a)^3 = -1$.

Option B is correct. The inverse of the matrix is calculated using the formula A^-1 = (1/det(A)) * adj(A).

Option B ($a,b,c$ are in GP) is correct because the determinant is zero for all values of $\alpha$ if and only if ${b^2} = ac$, which is the condition for a geometric progression.

Option B (A skew-symmetric matrix) is correct because the expression simplifies to a sum of odd powers of a skew-symmetric matrix, which is also skew-symmetric.

Option D (2) is correct because setting the determinant of the matrix to zero (since it's singular) and solving for $\lambda$ gives $\lambda = 2$.

Option D ($\left[\begin{array}{cc}1 & 2500 \\ 0 & 1\end{array}\right]$) is correct because the product of the matrices results in a matrix with the sum of odd numbers from 1 to 99 in the top right element, which equals 2500.

Option D (m x n) is correct because if AB' and B'A are defined, then the order of B must be m x n.

The correct answer is D (0) because when x=0, the determinant becomes a skew-symmetric determinant of odd order, which is always zero.

Option D (None of these) is correct because the given options do not match the calculated value of X^n for n=2.

Option D ($\lambda = 3,\mu \ne 10$) is correct because it satisfies the condition for no solution, where the determinant of the coefficient matrix is zero, but at least one of the determinants formed by replacing a column with the constants is non-zero.

Option A ((-6,11)) is correct because it satisfies the equation $6A^{-1} = A^2 + cA + dI$ when solving for c and d.

Option B is correct because the system has no solution when $\lambda = 3$ and $\mu \neq 10$, leading to an inconsistency in the equations.

Option A (-1) is correct because the determinant of the coefficient matrix must be zero for a non-trivial solution, leading to $\lambda = -1$.

Option A ($y(a-b)$) is correct because after applying column and row operations and using the given conditions, the determinant simplifies to $y(a-b)$.

Option D (4 x 3) is correct because A' is a 4x3 matrix, and for A'B and BA' to be defined, B must be a 3x4 matrix, making A'B a 4x4 matrix and BA' a 3x3 matrix. However, the question asks for the type of B, and since A'B is defined, B must have 3 rows, and since BA' is defined, B must have 4 columns.

Option D (None of these) is correct because (A+B)^-1 is a symmetric matrix, but it's not equal to A^-1 + B^-1.

Option C (0) is correct because the trace of a skew-symmetric matrix is always zero, as the diagonal elements of such a matrix are all zero.

Option B is correct because multiplying both sides of the equation AB = 3I by A⁻¹ and simplifying leads to A⁻¹ = (1/3)B.