Matrices practice · Easy

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because matrix addition is a binary operation and is commutative, but commutativity doesn't explain why it's a binary operation.

Option B (0) is correct because the determinant has a column of zeros after applying the column operation C1 -> C1 + C2 + C3, using the property 1 + w + w² = 0.

Option A (Null matrix) is correct because when $\alpha$ and $\beta$ differ by an odd multiple of $\pi/2$, the product E($\alpha$)E($\beta$) results in a null matrix.

Option A (Skew-symmetric matrix) is correct because if A is a skew-symmetric matrix of order 3, then A' = -A, and (A³)' = (A')³ = (-A)³ = -A³, which means A³ is also skew-symmetric.

Option D (None of these) is correct because the result depends on whether n is even or odd; A^n will be symmetric if n is even and skew-symmetric if n is odd, so it's not consistently one or the other.

Option B (1) is correct because the matrix is not invertible if its determinant is zero, and solving for 'a' when the determinant is zero gives a = 1.

Option D (A^n = nA - (n-1)I) is correct because it can be proven by mathematical induction that this formula holds for all n >= 1, given the matrix A.

Option A is correct because the adjugate of a 2x2 matrix is found by swapping the diagonal elements and changing the sign of the off-diagonal elements.

Option B ($81A$) is correct because calculating $A^2$ and then $A^3$ shows that $A^3 = 81A$.

Option C is correct. Given AB = A and BA = B, multiplying A by A results in A, and multiplying B by B results in B, indicating they are idempotent matrices.

Option B (916) is correct because it is obtained by solving the system of equations for infinite solutions and then substituting the values of lambda and mu into the given expression.

Option B ($ax(2a + 3x)$) is correct because after expanding the determinant and simplifying $f(2x) - f(x)$, the result matches this expression.

Option A ($I$) is correct because when matrix A and B are multiplied, the result is the identity matrix I.

Option D (skew-hermitian) is correct because the conjugate transpose of the matrix A is equal to -A.

Option D is correct because A²⁰¹¹ is equal to [[1, 0], [2011, 1]] when A = [[1, 0], [1, 1]], as shown by induction or pattern recognition.

Option C is correct. By finding matrix B and calculating B², we find that B² is the zero matrix.

Option A ($\frac{{13}}{{16}}$) is correct because it represents the probability that a 2x2 determinant with elements 0 and 1 is non-negative.

Option C (3) is correct because the sum of entries in each row of $A^n$ remains 1, and the sum of all entries in $A^n$ is equal to the order of $A$.

Option B (-4) is correct because substituting $\alpha = \sqrt[3]{3}$ into the determinant expression and simplifying yields -4.

Option A (contains two elements) is correct because the characteristic equation has two distinct roots (1 and -3), so the set of values of $\lambda$ contains two elements.

Option B ($1, -1$) is the correct answer because these are the values of $a$ that make the determinant of the coefficient matrix zero, ensuring non-zero solutions.

Option B ($|A|=\left|A^{\prime}\right|$) is correct because the determinant of a matrix is equal to the determinant of its transpose.

Option D (I) is correct because by using the given conditions and properties of matrix transpose and inverse, $B B^T$ simplifies to the identity matrix I.

Option B (1) is correct because there is only one value of k (k=1) for which the system has infinitely many solutions, which is found by setting the determinant to zero.

Option B ($A^2+B^2$) is correct because given $B = -A^{-1}BA$, it implies $AB + BA = 0$, and therefore $(A+B)^2 = A^2 + AB + BA + B^2 = A^2 + B^2$.

Option C, $A(- \theta)$, is correct because by finding the inverse of A and then multiplying by sec²θ, we obtain A(-θ).

Option D ($0,-(a+b+c)$) is correct because these are the values of x that make the determinant equal to zero.

Option A is correct because the determinant of the coefficient matrix is always positive, implying that the system has only the trivial solution for all values of $\lambda$ and $\theta$.

Option A (11) is correct. The absolute value of 5a+4b+3c+2d+e is found by first expanding the determinant, then substituting the coefficients into the expression.

Option D (2) is correct because setting A = A' leads to the equations a+1 = a^2-1 and 4a = a²⁺4, which are both satisfied when a = 2.