Matrices and Determinants

$$\mathrm{A}^{-1}=\frac{-1}{2}\left[\begin{array}{cc}1 & -4 \\ -1 & 2\end{array}\right]$$, then \(2 A+I_2=\quad\)
where \(I_2\) is a unit matrix of order 2

If $$A=\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right], \operatorname{adj} A=\left[\begin{array}{ccc}5 & x & -2 \\ 1 & 1 & 0 \\ -2 & -2 & y\end{array}\right]$$, then value of \(x+y\) is

If $$F(\propto)=\left[\begin{array}{ccc}\cos \propto & -\sin \propto & 0 \\ \sin \propto & \cos \propto & 0 \\ 0 & 0 & 1\end{array}\right]$$, where \(\propto \in R\), then \([F(\propto)]^{-1}=\)

If $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$$, then \(A^{-1}=\)

If $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1\end{array}\right]$$ and $$A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 c \\ 5 & -3 & 1\end{array}\right]$$, then values of a and c are respectively

If $$A=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$$, then \(\operatorname{adj} A=\)

If \(A = \left[ {\matrix{ 3 & 2 & 4 \cr 1 & 2 & 1 \cr 3 & 2 & 6 \cr } } \right]\) and A\(_{ij}\) are cofactors of the elements a\(_{ij}\) of A, then \({a_{11}}{A_{11}} + {a_{12}}{A_{12}} + {a_{13}}{A_{13}}\) is equal to

If inverse of $$\left[\begin{array}{ccc}1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6\end{array}\right]$$ does not exist, then \(x=\)

$$A(\propto)=\left[\begin{array}{cc}\cos \propto & \sin \propto \\ -\sin \propto & \cos \propto\end{array}\right]$$, then \(\left[A^2(\propto)\right]^{-1}=\)

If $$A^{-1}=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$$ and $$B^{-1}=\left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]$$, then \((A B)^{-1}=\)

If $$A^{-1}=\left[\begin{array}{lll}3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 5 & 5\end{array}\right]$$, then \(A=\)

The co-factors of the elements of second column of $$\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]$$ are

The cofactors of the elements of the first column of the matrix $A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 3 & 1 & 2 \\ -1 & 1 & 2\end{array}\right]$ are

If $A=\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]$ and $A^2-5 A-6 I=0$, then $A^{-1}=$

If $$A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$$ and $$A^{-1}=\left[\begin{array}{rrr}3 & -1 & 1 \\ \alpha & 6 & -5 \\ \beta & -2 & 2\end{array}\right]$$, then the values of \(\alpha\) and \(\beta\) ar...

The sum of the cofactors of the elements of second row of the matrix $$\left[\begin{array}{rrr}1 & 3 & 2 \\ -2 & 0 & 1 \\ 5 & 2 & 1\end{array}\right]$$ is

If $$A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], \quad B=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]$$, then \(B^{-1} A^{-1}=\)

The matrix $$A=\left[\begin{array}{rrr}a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2\end{array}\right]$$ is not invertible only if \(a=\)

If $A$ and $B$ are square matrices of order 3 such that $|A|=2,|B|=4$, then $|A(\operatorname{adj} B)|=\ldots$

If $A$ is non-singular matrix such that $(A-2 l)(A-4 I)=0$ then $A+8 A^{-1}=$ ..........

If $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$ and $A=A^{-1}$, then $x=\ldots \ldots$

If $A=\left[\begin{array}{cc}1+2 i & i \\ -i & 1-2 i\end{array}\right]$, where $i=\sqrt{-1}$, then $A(\operatorname{adj} A)=\ldots$

If $A$ is non-singular matrix and $(A+I)(A-I)=0$ then $A+A^{-1}=$ .............