Matrices practice · Mixed difficulty

Option D (10) is correct because the characteristic equation of the matrix A can be used to find the relationship between A² and A, which leads to the value of α + β.

Option B (2) is correct because after calculating $A^n$ and the determinant of $A^n - I$, we find that $\lambda$ equals 2.

Option B is correct because the inverse of the matrix $A{Q^{2021}}{A^T}$ simplifies to the inverse of $B^{2021}$, which is \beginpmatrix 1 & 0 -2021i & 1 \endpmatrix$\begin{{pmatrix}} 1 & 0 \\ -2021i & 1 \end{{pmatrix}}$.

Option A (Divisible by k) is correct because the determinant can be expressed as k times another determinant with integer elements.

Option C ($\pm \frac{{1}}{{\sqrt{{2}}}}, \pm \frac{{1}}{{\sqrt{{6}}}}, \pm \frac{{1}}{{\sqrt{{3}}}}$) is correct because the given matrix is orthogonal, implying that A*A^T = I, which leads to the specified values for a, b, and c after solving the resulting equations.

Option B (0) is correct because the determinant's first column becomes all zeros after applying the column operation, resulting in a zero determinant.

Option B ($\pm 1$) is correct because if $A{A^{T}} = I$, then $|A||{A^T}| = |I|$, which simplifies to $|A|^2 = 1$, thus $|A| = \pm 1$.

Option D (A-6I) is correct because it represents the expression for 10A⁻¹ in terms of A and the identity matrix I, derived by finding the inverse of A.

Option B (a skew symmetric matrix) is correct because the inverse of a skew-symmetric matrix is also skew-symmetric.

Option D is the correct answer because it represents the set of all 2x2 matrices that commute with the given matrix, derived by equating AB and BA and solving for the matrix elements.

The correct answer is A ($3(a+b+c)(b c+c a+a b)$) because by applying column and row operations, the determinant can be simplified to this expression.

Option D (None of the above) is correct because the inverse of (A+B) exists and is not a skew-symmetric matrix, nor is it equal to the sum of the inverses of A and B.

Option D (4) is correct because by finding the inverse of matrix A and comparing it with the given equation, we can determine the values of alpha and beta, and then calculate 4(alpha - beta).

Option A (0) is correct because a system of linear equations cannot have exactly two distinct solutions; it can have either zero, one, or infinitely many solutions.

Option B (3) is correct because, for the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero, leading to $\delta = -3$, and then solving for $k$ gives $k = 6$, so $\delta + k = 3$.

Option A (0) is correct because expanding the determinant and using the property $\omega^3 = 1$ results in the determinant being equal to zero.

Option D (-1,1) is correct because these values of k make the determinant of the coefficient matrix zero, leading to a non-zero solution.

Option A ($A$) is correct because simplifying the expression $A B+B A+I-(I-A)^2$ using the given conditions $A^2=A$ and $B=I-A$ results in $A$.

Option B ($\left[\begin{array}{cc}0 & 1 \\ 0 & 2+i\end{array}\right]$) is correct because it is the result of solving the given matrix equation for $X$.

Option D ($R-\left\{2,-3\right\}$) is correct because the system has only the trivial solution when the determinant of the coefficient matrix is non-zero, which occurs when $k$ is not equal to 2 or -3.

Option D (0) is correct because two columns become identical after simplification, making the determinant zero.

Option C (Any odd integer) is correct because the given condition simplifies to 1 + (-1)⁽ⁿ⁺²⁾ = 0, which is true only when n is an odd integer.

Option D is correct because applying the inverse properties to the expression results in C⁻¹BA⁻¹.

Option A (8) is correct because when the system has infinitely many solutions, the determinant of the coefficient matrix and related determinants must be zero, leading to $\beta - \alpha = 8$.

Option D (1) is correct because solving the system of equations for $m$ and $n$ and then substituting into $3m + 7n$ yields the value 1.

Option B ($\left( {4,3} \right)$) is correct because for this ordered pair, the determinant of the coefficient matrix is zero, and at least one of the determinants formed by replacing a column with the constants is non-zero, indicating an inconsistent system.

Option A ($\left[\begin{array}{cc}\sqrt{3} / 2 & 1 / 2 \\ -1 / 2 & \sqrt{3} / 2\end{array}\right]$) is correct because it represents the matrix A raised to the power of -50 when theta is pi/12, obtained by using the properties of rotation matrices.

Option A (square matrix) is correct because the matrix has the same number of rows and columns (3x3).

Option A ($|A| A$) is correct because the adjoint of the adjoint of a matrix $A$ is given by the formula $\operatorname{adj}(\operatorname{adj} A)=|A|^{n-2} A$, where $n$ is the order of the matrix. In this case, $n=3$.

Option B (0) is correct because the determinant simplifies to zero using trigonometric identities and the property A + B + C = π.