Consider the following statements in respect of two non-singular matrices A and B of the same order n:
1. \( adj(AB) = (adj A)(adj B) \)
2. \( adj(AB) = adj(BA) \)
3. \( (AB)adj(AB) - |AB|I_{n} \) is a null matrix of order n
How many of the above statements are correct?
Consider the following statements in respect of non-singular matrix A of order n:
1. \( A(adj A^{T}) = A(adj A)^{T} \)
2. If \( A^{2} = A \), then A is identity matrix of order n
3. If \( A^{3} = A \), then A is identity matrix of order n
Which of the statements given above are correct?
Consider the following statements in respect of a skew-symmetric matrix A of order 3:
1. All diagonal elements are zero.
2. The sum of all the diagonal elements of the matrix is zero.
3. A is orthogonal matrix.
Which of the statements given above are correct?
For what value of n is the determinant
\( \begin{vmatrix} C(9,4) & C(9,3) & C(10,n-2) \\ C(11,6) & C(11,5) & C(12,n) \\ C(m,7) & C(m,6) & C(m+1,n+1) \end{vmatrix} = 0 \)
for every m > n?
If ABC is a triangle, then what is the value of the determinant
\( \begin{vmatrix} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan(B + C) & \cos C \end{vmatrix} \)?
