Matrices

To find the product (AB), multiply the elements of the rows of (A) by the elements of the columns of (B) and add the products.

The determinant of a 2x2 matrix is calculated using the formula (ad - bc), where (a, b, c, d) are the elements of the matrix. In this case, the determinant of (C) is ((-2)(-1) - (3)(5) = 2 - 15 = -13).

The identity matrix of order 3 is a square matrix with 1s on the diagonal and 0s everywhere else.

If (A^2 = 0), then the determinant of (A) must be zero, as the determinant of a matrix is equal to the product of its eigenvalues.

The rank of a matrix is the maximum number of linearly independent rows or columns. In this case, the rows of (D) are linearly dependent, so the rank of (D) is 1.

A symmetric matrix is a square matrix that is equal to its transpose. In this case, only (\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}) is symmetric.

The trace of a matrix is the sum of its diagonal elements. In this case, the trace of (E) is (5 + (-1) = 4).

If (F^T = -F), then the determinant of (F) must be zero, as the determinant of a matrix is equal to the product of its eigenvalues.

An orthogonal matrix is a square matrix whose inverse is equal to its transpose. In this case, only (\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}) is orthogonal.

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector.

A diagonalizable matrix is a square matrix that can be expressed as a product of two matrices, one of which is a diagonal matrix. In this case, only (\begin{bmatrix} 1 & 0 \ 0 & 2 \end{bmatrix}) is diagonalizable.

The characteristic polynomial of a matrix is a polynomial whose roots are the eigenvalues of the matrix. In this case, the characteristic polynomial of (H) is (x^2 - 4x + 7).

A singular matrix is a square matrix whose determinant is zero. In this case, only (\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}) is singular.

The cofactor of an element (a_{ij}) of a matrix is the determinant of the submatrix obtained by deleting the (i)-th row and (j)-th column of the matrix, multiplied by ((-1)^{i+j}). In this case, the cofactor of (a_{23}) is ((-1)^{2+3}) times the determinant of the submatrix (\begin{bmatrix} 1 & 2 \ 7 & 8 \end{bmatrix}), which is (-9).