G Tewani Solutions for Chapter: Matrices, Exercise 3: DPP

If $A=\left[\begin{array}{cc}3-3& 4\\ 2-3& 4\\ 0-1& 1\end{array}\right]$, then the trace of the matrix $\text{adj}\left(\text{adj}A\right)$ is

If $A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 2& -3\\ 2& 1& 0\end{array}\right]$ and $B=\left(\text{adj}A\right)$ and $C=5A$, then find the value of $\frac{\left|\text{adj}B\right|}{\left|C\right|}$.

Let $A$ and $B$ be two non-singular square matrices, such that $B\ne I$ and $A{B}^2=BA$. If $A^3=B^{-1}{A}^3{B}^n$, then the value of $n$ is

If $A$ is an idempotent matrix satisfying $(I-0.4A{)}^{-1}=I-\alpha A$ (where $I$is the unit matrix of the same order as that of $A$), then the value of $\alpha$ is

If $A$ and $B$ are two non-singular matrices which commute, then ${\left(A(A+B{)}^{-1}B\right)}^{-1}\left(AB\right)$ is equal to

If $A$ and $B$ are non-singular matrices of order $n\times n$, such that $3AB{A}^{-1}+A=2A^{-1}BA$, then

If the matrices $A$ and $B$ are of $3\times 3$ and $(I-AB)$ is invertible, then which of the following statement(s) is/are correct?

If $A$ is a square matrix, such that $A.\left(\text{adj}A\right)=\left[\begin{array}{lll}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$, then