Determinant of Matrix

If the matrix A is such that [Tex]A = \begin{bmatrix}2 \\-4 \\7\end{bmatrix}\begin{bmatrix}1 & 9 & 5\end{bmatrix}[/Tex], then the determinant of A is equal to

The determinant of the matrix is 

Two eigenvalues of a 3 × 3 real matrix P are (2 + √ -1) and 3. The determinant of P is _____   

Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A⁻¹)^T is _________

Consider the following determinant:

[Tex]\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}[/Tex]

Which of the following is a factor of Δ?

The determinant of the matrix [Tex]\begin{bmatrix} 6 & -8 & 1 & 1 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & -1 \end{bmatrix}[/Tex] is:

The matrix A has (1, 2, 1)^T and (1, 1, 0)^T as eigenvectors, both with eigenvalue 7, and its trace is 2. The determinant of A is __________ .

Find the determinant of the following matrix:

[Tex]\begin{bmatrix} 6 & 0 & -1 & 2 \\ -1 & 2 & 3 & 6 \\ 4 & -3 & 0 & 0 \\ 1 & 5 & 7 & 2 \end{bmatrix}[/Tex]

Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB) = rank(A) × rank (B)
II. det(AB) = det(A) × det(B)
III. rank(A+B) ≤ rank(A) + rank(B)
IV. det(A+B) ≤ det(A) + det(B)

Which of the above statements are TRUE?

If the two matrices [Tex]\left[\begin{array}{lll} 1 & 0 & x \\ 0 & x & 1 \\ 0 & 1 & x \end{array}\right] and \left[\begin{array}{lll} x & 1 & 0 \\ x & 0 & 1 \\ 0 & x & 1 \end{array}\right][/Tex] have the same determinant, then the value of x is