Matrices practice · Easy

Option C (0) is correct because the determinant simplifies to zero after applying column operations, indicating linear dependence between columns.

The correct answer is C (Has infinitely many non-trivial solutions for $x=2z$) because the determinant of the coefficient matrix is zero, indicating non-trivial solutions, and the relationship $x=2z$ can be derived from the equations.

Option B (Invertible) is correct because the given condition and subsequent algebraic manipulation show that A+I has an inverse.

Option B ($x=\frac{1}{3}(a+b+c)$) is correct because the determinant vanishes when $x=\frac{1}{3}(a+b+c)$.

Option C ($m=n$) is correct because a square matrix is defined as a matrix with an equal number of rows and columns.

Option B (103) is correct because after calculating P⁵⁰ and subtracting I, the ratio (q31 + q32) / q21 equals 103.

Option D (${x^5}$) is correct because the determinant simplifies to ${x^4}({x^2} + 14)$, which is not divisible by ${x^5}$.

Option A ($\lambda = 5$) is correct because when $\lambda = 5$, the determinant of the coefficient matrix is zero, leading to infinitely many solutions for the homogeneous system.

Option D (None of these) is correct because the calculated value is 0, which is not among the given options.

Option A (- 12[ \beginarray*20c 4& - 2 - 3&1 \endarray ]$- \frac{1}{2}\left[ {\begin{{array}}{{{*{20}{c}}}} 4&{ - 2}\\ { - 3}&1 \end{{array}}} \right]$) is correct because it is the correct inverse matrix calculated using the formula for a 2x2 matrix inverse.

Option C $\left[\begin{array}{lll}8 & 0 & 0 \\0 & 8 & 0 \\0 & 0 & 8\end{array}\right]$ is correct because $A(\operatorname{Adj} A) = |A|I$, and in this case, $|A| = 8$.

Option D (none of these) is correct. The determinant simplifies to 0, which is not equal to any of the given options.

Option B ($|A B|=0$) is correct because the determinant of a skew-symmetric matrix of odd order is always 0, and $|A B|=|A||B|$. Since $|B|=0$, $|A B|=0$.

Option D (any odd integer) is correct because the equation simplifies to $2\left| {\begin{array}{cccccccccccccccccccc} a&{a + 1}&{a - 1} \\ { - b}&{b + 1}&{b - 1} \\ c&{c - 1}&{c + 1} \end{array}} \right| = 0$ when n is odd.

Option A ($\begin{bmatrix} 1&0\\0&1 \end{bmatrix}$) is correct because ${A^2}$ equals the identity matrix, and therefore ${A^4}$ also equals the identity matrix.

Option B (It is associative) is correct because matrix multiplication is associative, meaning (AB)C = A(BC).

Option A (Every identity matrix is a scalar matrix) is correct because an identity matrix is a special case of a scalar matrix where the scalar is 1.

Option C ([z]) is correct because substituting the greatest integer values for x, y, and z into the determinant results in a value of 1, which equals [z].

Option D, 36, is correct because the number of such matrices A is equal to the number of positive integer solutions to x + y + z = 10, which is given by $^{9}C_2 = 36$.

Option C (-2-$\sqrt{3}$) is correct because solving the determinant equation leads to tan(x) = 1, and substituting into the expression yields -2-$\sqrt{3}$.

Option A (1) is correct because after row operations, the matrix reduces to a single non-zero row, indicating a rank of 1.

Option A (-2) is correct because after applying row operations R2 -> R2 - R1 and R3 -> R3 - R1 and expanding along C3, the determinant simplifies to -2.

Option B is correct. The inverse of a diagonal matrix is a diagonal matrix with the reciprocals of the original diagonal elements, and their sum is 11/6.

Option C (-1) is correct because by comparing the degree of the determinant and the RHS, we find that n = -1.

Option B ($a = 1$) is correct because a matrix is singular if its determinant is zero, and setting the determinant of the given matrix to zero yields $a = 1$.

Option C ($B^5-A^5$ is a skew-symmetric matrix) is NOT true because the transpose of $B^5 - A^5$ is not equal to its negative, hence it is not skew-symmetric.

Option B ($\frac{{1}}{{\mathrm{det\\;}\left( A \right)}}$) is correct because the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.

Option D ($|A+A|=2|A|$) is correct because $|A+A| = |2A| = 2^3|A| = 8|A|$ for a 3x3 matrix A, not $2|A|$.

Option A ($A$) is correct because $A = \omega I_2$, and since $\omega^3 = 1$, $A^{100} = (\omega^{100})I_2 = \omega I_2 = A$.

Option D ($x=0$) is correct because $x=0$ satisfies the given determinant equation after simplification using properties of cube roots of unity.