Let $A=\left[\begin{array}{ccc}1 & 4 & 2 \\ 2 & -1 & 4 \\ -3 & 7 & -6\end{array}\right]$ and $B=\left[b_{i j}\right]_{3 \times 3}$ with $b_{11}=2$, $b_{13}=-2, b_{12}=0$ is such that $A B=\left[\begin{array}{ccc}2 & 14 & -4 \\ 4 & 1 & -8 \\ -6 & 15 & 12\end{array}\right]$, then $|B|+\operatorname{trace}(B)=$
A is a $m \times n$ matrix of rank 4 . If A contains an $m$ th order non singular sub matrix and $A^T A$ is a $7 \times 7$ matrix, then the number of rows of $A$ is
If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real number $\mathbf{R}$ such that $C D=-D C$, then $n$ is
If $A, B$ are two non singular matrices of order $3,|B|=k$, a positive integer, then match the items of list-I with the items of list-II.
| $$ \text { List-I } $$ |
$$ \text { List-II } $$ |
||
|---|---|---|---|
| A. | $\quad\left|k^{-1} A^{-1}\right|$ | I. | $$ B A^k+A^k B $$ |
| B. | $\left|\operatorname{Adj}\left(A^{-1}\right)\right|$ | II. | $$ \frac{B \operatorname{Adj}(B)}{|B|} $$ |
| C. | $B A B^{-1}=I, \Rightarrow B A^k B^{-1}=$ | III. | $$ \frac{1}{|B|^3|A|} $$ |
| D. | $\quad \operatorname{Adj}\left(\operatorname{Adj}\left(A^{-1}\right)\right)=$ | IV. | $$ \frac{1}{|A|}\left(A^{-1}\right) $$ |
| V. | $$ \frac{1}{|A|^2} $$ |
||
$$ \text { The correct match is } $$
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