Matrices practice · Medium

Option C (20) is correct because after calculating the determinant, the resulting integer (6140) is exactly divisible by 20.

Option B (S-I True, S-II True) is correct because $f(x) \cdot f(y) = f(x+y)$ and $f(-x)$ is invertible since its determinant is non-zero.

Option B ([ \beginarraycc 1 & 4032 0 & 1 \endarray ]$\left[ {\begin{{array}}{{cc}} 1 & 4032 \\ 0 & 1 \end{{array}}} \right]$) is correct because after simplifying the matrix expression, the result is a matrix with 1, 4032, 0, and 1 as its elements.

The correct answer is B ($k=80$ if $A$ is non skew symmetric matrix) because the total number of matrices with the given condition is 85, and the number of skew-symmetric matrices is 5, so the number of non-skew-symmetric matrices is 80.

Option A (3) is correct because substituting n=3 into the row operation yields the given transformed matrix.

Option B (2) is correct because by applying column operations to the left-hand side determinant, it simplifies to 2 times the determinant on the right-hand side, thus k=2.

Option C (0) is correct because the sum of the determinants results in a determinant with two identical columns, making its value zero.

Option C ($\lambda=-5$) is correct because it satisfies the condition for non-trivial solutions of the system of equations.

Option A (1/9) is correct because it is the value of |A| calculated using the given relationship between the elements of matrices A and B and their determinants.

Option B is correct. The expression (I+A)(I-A)^(-1) simplifies to the matrix [[0, 1], [-1, -1]] after matrix addition, subtraction, and inversion.

Option D ($|A|$) is correct because the trace of a skew-symmetric matrix is always 0, and the determinant of a 3x3 skew-symmetric matrix is also 0.

Option D (None of these) is incorrect. The determinant A is independent of $\theta$ and its value is 2.

Option A ($a_1 b_2 c_3$) is correct because after determinant operations, the ratio simplifies to this product.

Option B ($(2006)^2$) is correct because the determinant of $A_r$ simplifies to $2r-1$, and the sum of $2r-1$ from r=1 to 2006 is $(2006)^2$.

Option D (20) is correct because by counting the total number of matrices and subtracting the singular ones, we find 20 non-singular matrices.

Option D (5) is correct because the system has non-trivial solutions only if the determinant of the coefficient matrix is zero, leading to λ = 5.

Option D (-3 w²) is correct because after applying column operations and using the property 1 + w + w² = 0, the determinant simplifies to -3 w².

Option A ($\log_e^2$) is correct because the limit of the Riemann sum can be expressed as a definite integral, which evaluates to ln(2).

Option C (4) is correct because by using the given conditions $A A^T = 16I$ and $abc=2$, we can find the value of $(a^3+b^3+c^3)^2$.

Option B ($x= \pm 1 / \sqrt{2}, y= \pm 1 / \sqrt{6}, z= \pm 1 / \sqrt{3}$) is correct because these values satisfy the condition $A' = A^{-1}$, which implies $AA' = I$.

Option D ($2(a + b + c)^3$) is correct because applying column and row operations simplifies the determinant to $2(a + b + c)^3$.

The correct answer is D ($- \frac{1}{2}$). The integral of the determinant, after simplification, evaluates to -1/2.

Option C ([\beginarraycc1 & 2015 0 & 1\endarray]$\left[\begin{{array}}{{cc}}1 & 2015 \\ 0 & 1\end{{array}}\right]$) is correct because after simplifying the expression $P^T Q^{2015} P$, it reduces to $A^{2015}$, which can be easily calculated.

Option C (At most 2) is correct because the determinant is a polynomial of degree at most 2 after simplification.

Option C (An odd multiple of $\pi /2$) is correct because the product AB being a null matrix implies that $\cos(\alpha - \beta) = 0$, which means $\alpha - \beta$ is an odd multiple of $\pi /2$.

Option A (I) is correct because when the given matrix A is squared, the result is the identity matrix I.

Option C ($2(10!11!12!)$) is correct because the determinant simplifies to this value after factoring and applying row operations.

Option B ($x=\left[\begin{array}{ccc}2 & -1 & 1 \\ 1 & 2 & 0\end{array}\right]$) is correct because solving the system of matrix equations yields this result for matrix x.

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

Option C ($\frac{3}{8}$) is correct because it is the probability of exactly two missiles hitting the target, considering the probability of being destroyed and the probability of hitting the target if not destroyed.