Matrices practice · Mixed difficulty

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 B (infinite number of solutions) is correct because the system of homogeneous equations always has at least the trivial solution (x=y=z=0), and if the determinant of the coefficient matrix is zero, there are infinitely many solutions.

Option C (Has infinitely many solutions for $\alpha=-1$) is correct because when $\alpha=-1$, the determinant of the coefficient matrix is zero, and the system has infinitely many solutions if the equations are consistent.

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 A ($\det A$) is correct because the determinant of $A^{-1} \text{adj} A$ simplifies to $\det A$ using determinant properties.

Option C ((-3,-1)) is correct because after simplifying the determinant using column and row operations, the expression becomes $-(2 + \sin 2x)$, and the range of this expression is [-3, -1].

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 ($a = 1$) 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.

Option C (Both I and II are true) is correct because a matrix is an arrangement of numbers, while a determinant can be reduced to a single number.

Option A (0) is correct because the sum of the real roots of the given cubic equation, obtained by expanding the determinant, is 0.

Option C (${14^4}$) is the correct answer because it follows from the properties of adjoints and determinants of matrices.

Option A (Only I is true) is correct because if A=B, then multiplying both sides by C will result in AC=BC. However, the converse is not always true.

Option C (Negative) is correct because the determinant simplifies to an expression that is negative given the conditions $b^2 - ac < 0$ and $a > 0$.

Option A (Only I is true) is correct because a scalar matrix with k=1 is an identity matrix, but not every identity matrix is *not* a scalar matrix.

Option A is correct because the assertion is true, the reason is true, and the reason correctly explains the assertion about diagonal matrices.

Option A ($A^{2} B=B A^{2}$) is correct because it is derived from the given equation by algebraic manipulation and using the properties of matrices.

Option B ($\frac{{72}}{{5}},-\frac{{21}}{{5}}$) is correct because for infinitely many solutions, the determinant of the coefficient matrix and the determinant with the z-column replaced must both be zero.

Option A ($A_{\alpha+\beta}$) is correct because the product of the two rotation matrices results in another rotation matrix with an angle equal to the sum of the individual angles.

Option D (${C^{ - 1}}B{A^{ - 1}}$) is correct because it follows the rule of inverting a product of matrices: (ABC)^-1 = C^-1 B^-1 A^-1.

Option A (-5) is correct because the minimum value of det(A) is found by maximizing sin²(θ), which leads to d = -5.

Option C (A scalar matrix) is correct because multiplying a unit matrix by a scalar results in a scalar matrix.

Option D ([ \beginarray*20c 8&4 8&0 \endarray ]$\left[ {\begin{{array}}{{{*{20}{c}}}} 8&4\\ 8&0 \end{{array}}} \right]$) is correct because it is the result of substituting matrix A into the given polynomial function and performing the matrix operations.