The determinant of J is calculated as 1*(1*1 - 0*3) - 2*(0*1 - 0*2) + 1*(0*3 - 1*2) = 1 - 0 - 2 = -1.

The eigenvalues are found by solving the characteristic equation det(G - λI) = 0. This gives λ^2 - 8λ + 7 = 0, which factors to (λ - 1)(λ - 7) = 0, hence λ = 1, 7.

The order of the product of two matrices is determined by the outer dimensions. Here, A (2x2) and B (2x3) can be multiplied, resulting in a matrix of order 2x3.

A 3x3 matrix has 3 rows and 3 columns, resulting in a total of 3 * 3 = 9 elements.

In a diagonal matrix, all non-diagonal elements are zero.

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

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

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

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

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

A triangle with sides 3 cm, 4 cm, and 5 cm satisfies the Pythagorean theorem (3² + 4² = 5²), thus it is a right triangle.

The area of a circle is given by A = πr². If the radius is doubled (r becomes 2r), the new area is A' = π(2r)² = 4πr², which is four times the original area.

In a parallelogram, opposite angles are equal. Therefore, if one angle is 70 degrees, the opposite angle is also 70 degrees.

In a 30-60-90 triangle, the sides opposite the 30 degrees and 60 degrees angles are in the ratio 1:√3.

The area of a triangle is given by A = 1/2 * base * height. Here, A = 1/2 * 8 * 5 = 20 cm².

The determinant of a 2x2 matrix is calculated as ad - bc.

The length of the diagonal d of a rectangle can be found using the Pythagorean theorem: d = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

The order of a matrix is given by the number of rows followed by the number of columns. Therefore, a matrix with 3 rows and 4 columns is of order 3x4.

The perimeter of a rectangle is given by P = 2(length + width). Here, P = 2(10 + 5) = 30 cm.

The rank of E is 2 because the rows are linearly dependent (the third row is a linear combination of the first two).

The trace of a square matrix is defined as the sum of the elements on its main diagonal.

The trace of a matrix is the sum of its diagonal elements. Here, trace(H) = 2 + 3 + 4 = 9.

A matrix where all elements are zero is called a zero matrix.

A square matrix is one where the number of rows is equal to the number of columns. Among the options, only 3x3 has equal rows and columns.

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. The matrix [[1, 0], [0, 1]] fits this definition.

A symmetric matrix is one that is equal to its transpose. Matrix B is symmetric because B = B^T.