A symmetric matrix satisfies the condition Aij = Aji for all i and j.

A symmetric matrix is defined by the property A = A^T, meaning it is equal to its transpose.

A matrix A is symmetric if it is equal to its transpose, i.e., A = A^T.

The perimeter P of a square is given by P = 4 * side. If P = 32 cm, then side = 32/4 = 8 cm.

The rank of matrix B is the number of non-zero rows in its row echelon form, which is 3.

The determinant of B is calculated as (2*7) - (3*5) = 14 - 15 = -1.

The determinant of C is 0 because it has a row of zeros.

Using the determinant formula for 3x3 matrices, det(C) = 1(3*0 - 1*1) - 0 + 2(-1*1 - 3*2) = 0 - 0 - 12 = -12.

The determinant of an upper triangular matrix is the product of its diagonal elements. Here, det(C) = 1 * 1 * 1 = 1.

C^2 = C * C = [[1*1 + 2*3, 1*2 + 2*5], [3*1 + 5*3, 3*2 + 5*5]] = [[11, 16], [18, 35]].

The trace of a matrix is the sum of its diagonal elements. Trace(D) = 2 + 2 = 4.

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10, and adj(D) = [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

Matrix E is singular because its determinant is 0 (1*4 - 2*2 = 0).

The inverse of the identity matrix F is itself, so F^(-1) = F.

The determinant of F is calculated as (1*4) - (2*2) = 4 - 4 = 0.

The second row is a multiple of the first row, so there is only one linearly independent row. Therefore, the rank of F is 1.

The trace of F is the sum of the diagonal elements: 1 + 5 = 6.

The eigenvalues are found by solving the characteristic equation det(G - λI) = 0. The eigenvalues are λ = 1, 8.

The determinant of H is calculated as (0*-0) - (1*-1) = 0 + 1 = 1.

Using the determinant formula for 3x3 matrices, det(H) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = 0 - 40 - 15 = -55.

The rank of H is 1 because the second row is a multiple of the first row.

The eigenvalues are found by solving the characteristic equation: det(H - λI) = 0, which gives λ^2 - 9λ + 1 = 0.

The rank of I is 1 because all rows are linearly dependent.

Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = 0 - 40 - 15 = -55.

The trace of a matrix is the sum of its diagonal elements. Thus, trace(I) = 2 + 2 = 4.

The rank of J is 1 because both rows are linearly dependent (they are identical).

The determinant of J is 0, as the rows are linearly dependent.

The circumference of a circle is given by C = πd, where d is the diameter. Here, C = π * 14 cm ≈ 44 cm.

The volume of a sphere is given by V = (4/3)πr³. The radius r = diameter/2 = 5 cm. Thus, V = (4/3)π(5)³ = (4/3)π(125) = 500/3 π cm³.