Let A be a nonsingular diagonalizable matrix of order 3 with eigenvalues λ1, λ2, λ3. Then A-1 is diagonalizable if:
λ1 = 2, λ2 = 1, λ3 = 3
The inverse of every nonsingular diagonalizable matrix is diagonalizable. Only the listed triple (2,1,3) has no zero eigenvalue and is consistent with nonsingularity. The other triples make the matrix singular.
Important Questions from Determinants
- An $n \times n$ square matrix $A$ satisfies $A^T = A^{-1}$. The determinant of this matrix may take which of the following value(s)?
The determinant of the matrix $\begin{pmatrix} 1 & p & q + r \\ 1 & q & r + p \\ 1 & r & p + q \end{pmatrix}$ is ______.
Let $f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}$
where $p$ is a constant.
The value of $\frac{d^3}{dx^3}f(x)$ at $x = 0$ is
How many matrices of different orders are possible with elements comprising all prime numbers less than 30?
If A is a square matrix of order 3 and det A = 5, then what is det [(2A) -1 ] equal to ?
