For what value of k is the matrix \(\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix}\)  singular?

Let A be a skew-symmetric matrix of order 3.

What is the value of det(4A⁴) - det(3A³) + det(2A²) - det(A) + det(-I)  where I is the identity matrix of order 3?

Let A and B be symmetric matrices of same order, then which one of the following is correct regarding (AB - BA) ?

1. Its diagonal entries are equal but nonzero

2. The sum of its non-diagonal entries is zero

Select the correct answer using the code given below :

If \[ [\,x \;\; 1 \;\; 1\,] \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} = \begin{bmatrix} 45 \end{bmatrix} \] then which one of the following is a value of \(x\)? 

If \[ A = \begin{bmatrix} y & z & x \\ z & x & y \\ x & y & z \end{bmatrix} \] where \( x, y, z \) are integers, is an orthogonal matrix, then what is the value of \( x^2 + y^2 + z^2 \)?

Consider the following in respect of a non-singular matrix \( M \):

1. \( |M^2| = |M|^2 \)
2. \( |M| = |M^{-1}| \)
3. \( |M| = |M^T| \)

How many of the above are correct?

If \[ f(\theta) = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \] then what is \( f(\pi)^2 \) equal to?

A generic 3 × 3 real matrix A has eigenvalues 0, 1 and 6, and I is the 3 × 3 identity matrix. The quantity/quantities that cannot be determined from this information is/are the

If \(A = \(\left[ {\begin{array}{} {coshx}&{sinhx}\\ { - sinhx}&{coshx} \end{array}} \right])\) , then trace (A ²) is equal to

Let A be a non-singular diagonalisable matrix of order 3 with eignvalues λ1, λ2, λ3. A ^-1 is diagonalisable if:

Assertion(A): If A is any Matrix given by A = \(\left(\begin{array}{ccc}5 & 0 & 3 \\ −1 & 0 & 2 \\ 1 & 0 & 1\end{array}\right)\)

Then det A = 0, since all elements in column II are zero

Reason (R): Laplace expansion permits evaluation of a determinant along any row or column