Matrices practice · Mixed difficulty

The correct answer is C. The sum of the infinite matrix series can be found by using the formula for the sum of an infinite geometric series and then inverting the resulting matrix.

Option A (1023) is correct because by repeatedly applying the given condition $A^2B = BA$, we can derive the formula $(AB)^n = A^{2^n-1}B^n$, and thus find $k = 2^{10} - 1 = 1023$.

Option C (0) is correct because the determinant simplifies to zero due to the properties of arithmetic progressions and determinant operations.

Option C (4) is correct because after applying row and column operations and simplifying, the determinant equals $4a^2b^2c^2$, thus k = 4.

Option B (0) is correct because the determinant is zero due to two rows (or columns after taking common factors) being identical.

Option B is correct. The adjoint of the given matrix is found by calculating the cofactors, forming the cofactor matrix, and then taking the transpose.

Option C is correct. A² is found by multiplying matrix A by itself, and using trigonometric identities simplifies the result to the given matrix.

Option D (-25) is correct. By calculating A² - 2A - I and then finding its determinant, and using the property |A^n| = |A|^n, the determinant of the given matrix is -25.

Option C (4) is correct because $\left|\left(\operatorname{adj} A^{-1}\right)^{-1}\right|$ simplifies to $|A|^2$, and since $|A|=2$, the result is 4.

Option A (I is true and II is false) is correct because all three matrices are diagonal, but their orders are 1, 2, and 3, respectively.

Option A ([ \beginarray*20c 1 2005 0 1 \endarray ]$\left[ {\begin{{array}}{*{20}{c}} {1\,\,\,\,2005}\\ {0\,\,\,\,\,\,\,\,1} \end{{array}}} \right]$) is correct because it follows from simplifying the expression and using the properties of matrix powers.

Option D (Does not exist) is correct because a skew-symmetric matrix of odd order has a determinant of 0, making it non-invertible.

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

Option C (-I) is correct because when A' is multiplied by A, the result is a negative identity matrix.

Option B (5) is correct because for a symmetric matrix, the elements Aij and Aji must be equal, leading to the equation x + 2 = 2x - 3, which solves to x = 5.

Option B (0) is correct because when two columns of A are interchanged to form B, det(B) = -det(A), and therefore det(A+B) = 0.

Option B ($abc$) is correct because given the condition ${a^{ - 1}} + {b^{ - 1}} + {c^{ - 1}} = 0$, the determinant simplifies to abc.

Option A (0) is correct because after taking common factors and applying row operations, the determinant simplifies to zero.

Option A (5) is correct because by equating the corresponding elements of the matrices, we find $x=1$, $y=-2$ (since $y<0$), and $z=2$. Therefore, $x-y+z = 1 - (-2) + 2 = 5$.

Option B ($\left| A \right|^{n - 1}$) is correct because the determinant of the adjoint of a matrix A is equal to the determinant of A raised to the power of (n-1), where n is the order of the matrix.

Option D (0) is correct because the determinant is zero due to the properties of cube roots of unity.

Option A [ \beginarray20c 1&3 2& - 1 \endarray ]$\left[ {\begin{{array}}{{{20}{c}}} 1&3\\\\2&{ - 1} \end{{array}}} \right]$ is correct because it is the matrix X that satisfies the given matrix equation.

Option D (-2) is correct because when alpha is -2, the determinant of the coefficient matrix is zero, but at least one of the determinants formed by replacing a column with the constants is non-zero, indicating no solution.

Option A (${A^{ - 1}}$) is correct because the determinant of A is 1, making the adjugate of A equal to its inverse.

Option B (0) is correct because the expression simplifies to zero due to the properties of the commutator and matrix algebra.

Option A (0) is correct because three planes cannot intersect at exactly two distinct points; the intersection would either be a line or no intersection.

Option A (0) is correct because when n = 3k, the determinant simplifies to zero due to the properties of cube roots of unity.

Option D (-2, -1) is correct because, after substituting the matrix A into the equation AA^T = 9I and solving the resulting system of equations, the ordered pair (a, b) is found to be (-2, -1).

The correct answer is B ($\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P) because the determinant of the coefficient matrix must be zero for a non-zero solution, leading to the given relationship between a, b, and c.

Option A (1) is correct because the determinant of the coefficient matrix must be zero for non-zero solutions, leading to the given equation equaling 1.