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.

Det(I) = (3*4) - (2*1) = 12 - 2 = 10.

Det(D) = (1*4) - (2*3) = 4 - 6 = -2.

The characteristic polynomial is det(G - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0. The eigenvalues are λ = 1 and λ = 3.

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 inverse of D is (1/det(D)) * adj(D) = (1/(4*6 - 7*2)) * [[6, -7], [-2, 4]] = [[3/2, -7/4], [-1/2, 2/4]].

The inverse of F is given by (1/det(F)) * adj(F). Here, det(F) = 4*6 - 7*2 = 10, and adj(F) = [[6, -7], [-2, 4]]. Thus, F^(-1) = (1/10) * [[6, -7], [-2, 4]] = [[3/5, -7/10], [-1/5, 2/5]].

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]].

Calculating J^2 gives [[0, 1], [1, 0]] * [[0, 1], [1, 0]] = [[1, 0], [0, 1]], which is the identity matrix.

Using the determinant formula, det(J) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) + 3*(-5) = -24 + 40 - 15 = 1.

Det(F) = (2*4) - (1*3) = 8 - 3 = 5.

A^2 = A * A = [[2*2 + 3*1, 2*3 + 3*4], [1*2 + 4*1, 1*3 + 4*4]] = [[10, 21], [21, 16]].

The inverse of A is given by (1/det(A)) * adj(A). Det(A) = (2*4) - (3*1) = 5. The adjoint is [[4, -3], [-1, 2]]. Thus, A^(-1) = (1/5) * [[4, -3], [-1, 2]].

The order of the product of two matrices is determined by the outer dimensions. Since both A and B are 2x2 matrices, their product AB will also be a 2x2 matrix.

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.

The maximum number of non-zero elements in the sum of two matrices occurs when all elements of both matrices are non-zero. Therefore, A + B can have a maximum of 9 non-zero elements.

The sum of two matrices of the same order results in a matrix of the same order. Therefore, A + B will be a 3x3 matrix.

The order of the resultant matrix when two matrices are multiplied is determined by the outer dimensions. Here, both A and B are 3x3, so the product AB is also 3x3.

The product of two matrices A and B, both of order 3x3, will also be a matrix of order 3x3.

The order of the sum of two matrices is the same as the order of the individual matrices. Therefore, A + B is a 3x3 matrix.

The sum of two matrices is defined only when they have the same order. Since both A and B are 3x3 matrices, A + B will also be a 3x3 matrix.

The product of two matrices A and B, both of order 3x3, will also be a matrix of order 3x3.

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

A 3x3 matrix can have a maximum of 3 linearly independent rows, but the answer options are incorrect. The correct answer is 3.

A matrix with more columns than rows is referred to as a rectangular matrix.

A matrix that is both upper and lower triangular must have non-zero elements only on the diagonal, making it a diagonal matrix.

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

A diagonal matrix has non-zero elements only on its main diagonal, while all other elements are zero.

An orthogonal matrix is defined as a square matrix whose transpose is equal to its inverse.

A skew-symmetric matrix satisfies the condition a_ij = -a_ji for all i and j.