Matrices practice · Easy

Option C (Skew symmetrix matrix) is correct because if A+Aᵀ is symmetric, then A-Aᵀ is skew-symmetric.

Option C (1) is correct because the given equations imply that the determinant of the coefficient matrix is zero, leading to $a^2+b^2+c^2+2abc = 1$.

Option D ($\pi$) is correct because by applying trigonometric identities and L'Hôpital's rule, the limit simplifies to $\pi$.

Option B ($a=-\frac{{3}}{{2}}, b \neq 0, c=\frac{{1}}{{2}}$) is correct because for f to be continuous at x=0, the left-hand limit, right-hand limit, and the value of the function at x=0 must be equal, leading to these specific values for a and c.

Option D (equal to R-{0}) is correct because the system has a unique solution when the determinant of the coefficient matrix is non-zero, which occurs when k is not equal to 0.

Option B (1) is correct because the determinant of any identity matrix, regardless of its order, is always 1.

Option A (2,7) is correct because these are the other two roots of the cubic equation obtained from the determinant.

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

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

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 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 D (None of these) is correct because none of the given options satisfy the condition $(I-A)^{-1}$ when $A^5=O$ and $A^n \neq I$ for $1 \leq n \leq 4$.

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 (100) is correct because the sum of all elements of the matrix $\sum\limits_{{n=1}}^{50} B^{n}$ is calculated to be 100.

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 $A^2 + B^2$.

Option A (3) is correct because for a matrix to be singular, its determinant must be zero. Solving the determinant equation yields $\lambda = 3$.

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.

The correct answer is B ($5\sqrt 3 \left( {\sqrt 6 - 5} \right)$) because after splitting the determinant and simplifying, the value is obtained.

Option A ($\alpha=a^2+b^2, \beta=2 a b$) is correct because it results from performing the matrix multiplication $A^2$ and equating the elements.

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

Option C (0) is correct because after applying row and column operations, two columns become identical, resulting in a determinant of zero.

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

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 B ($1-\alpha^2-\beta \gamma=0$) is correct because it is the result of the matrix multiplication and equating it to the identity matrix.

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