Matrices practice · Mixed difficulty

Option B (If ${A^3} + 2{A^2} + 3A + 5I = O;\mathrm{then}\;A\;\mathrm{is\;invertible}$) is correct because the equation can be rearranged to show that A has an inverse.

Option B (1) is correct because substituting x = 1 into the determinant equation results in 0, indicating that 1 is a root.

Option D ($ab = 1$) is correct because after calculating the fourth power of the matrix and equating it to the identity matrix, we find that $ab = 1$.

Option C (1, -4) is correct because by finding the values of p, c, b, and q from the given roots, we can form the equation $x^2 + bx + c = 0$ and solve for its roots.

Option D $\left( {3,\frac{{1}}{{81}}} \right)$ is correct because by using the properties of adjoint and determinant of a matrix, we can find the values of |λ| and μ.

Option A is correct. For a 2x2 matrix, adj(adj A) = A and |adj A| = |A|, but the second statement doesn't explain the first.

Option D (5) is correct. By using the property $A(adj A) = |A|I = AA^T$, we can deduce that $adj A = A^T$. Comparing the elements of $adj A$ and $A^T$ allows us to solve for $a$ and $b$, and then calculate $5a + b$.

Option A (-1) is correct because the cofactor is the negative of the minor of the element -3, which is in the 3rd row and 2nd column.

Option C (null matrix) is correct because the adjoint of the transpose of a matrix is the transpose of the adjoint of the matrix, hence their difference is a null matrix.

Option B (Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1) is correct because both statements are true based on properties of adjoints, but one doesn't explain the other.

Option D (${A^n} = nA - (n - 1)I$) is correct because it holds true for n=1, n=2, and n=3, satisfying the principle of mathematical induction.

Option C ($2\sqrt 3$) is correct because the minimum value of the expression is found using AM-GM inequality, resulting in $2\sqrt 3$.

Statement-1 is true because A(BA) and (AB)A are symmetric matrices due to the associative property and symmetry of A and B. Statement-2 is also true because if AB = BA, then AB is symmetric. However, statement-2 is not a correct explanation for statement-1.

Option A (8) is correct because it is the result of calculating the ratio of |adj B| to |C| using properties of adjoint and determinants.

The correct answer is none of these because the given condition $A(AB) = B(BA)$ does not necessarily imply $A^2 + B^2 = 0$ or $I$.

Option D (Skew symmetric matrix) is correct because the given matrix A, where $a_{ij} = i^2 - j^2$, satisfies the condition for a skew-symmetric matrix, i.e., $a_{ij} = -a_{ji}$.

Option A (-5) is the correct value of k because it makes the system of linear equations have infinitely many solutions by making the equations linearly dependent.

Option B (-4) is correct because substituting the value of alpha into the function and simplifying yields -4.

Option A (0) is correct because since $x, y, z$ are in AP, $2y = x + z$. Applying row operations makes the second row zero, resulting in a zero determinant.

Option C (5) is correct because it satisfies the condition that B is the inverse of A, leading to AB = I, and solving for alpha.

Option D (81) is correct because |adj A| = |A|^(n-1), where n is the order of the matrix, and |A| = 9.

Option B ($I$) is correct because after simplification, $X$ equals $I$, and the inverse of the identity matrix is itself.

The correct answer is B ($\left|\begin{array}{cc}a x+b y & b x+c y \\ a^{\prime} x+b^{\prime} y & b^{\prime} x+c^{\prime} y\end{array}\right|$) because by applying column operations, the determinant can be reduced to a 2x2 determinant.

Option D (None of these) is correct because the matrix multiplication AB results in a 1x1 matrix [5], which is not any of the given options.

Option A ($1,3,6$) is correct because the characteristic roots (eigenvalues) of a triangular matrix are its diagonal elements.

Option A ($A^2$) is correct because if $A^3 = I$, multiplying both sides by $A^{-1}$ gives $A^2 = A^{-1}$.

Option B (purely imaginary) is correct because the determinant is shown to be equal to the negative of its conjugate.

Option C (orthogonal) is correct because the matrix satisfies the condition A*A' = I, where A' is the transpose of A and I is the identity matrix.

The correct answer is that there exist infinitely many $B$'s such that $AB = BA$ because $AB = BA$ only when $a = b$, and $a, b \in N$.

Option B ((-2,-1)) is correct because when A*A^T = 9I, solving the resulting equations gives a = -2 and b = -1.