Adjoint of a matrix

Q.

If the matrices $A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right]$, $B=adjA$, $C=3A$ then, $\frac{\left|adjB\right|}{\left|C\right|}$ is equal to

1. $16$

2. $2$

3. $8$

4. $72$

Q. which of the following is true, if A is an invertible square matrix :-

1. ${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T$
2. ${\left(A^T\right)}^{-1}=\left(A^{-1}\right)\left(A^T\right)$
3. $\left(A^T\right)={\left(A^{-1}\right)}^T$
4. ${\left(A^T\right)}^{-1}=\left(A^T\right)\left(A^{-1}\right)$

Q. If A is a square matrix of order 3 such that $\left|A\right|=2$ then $\left|\right(adj{A}^{-1}{)}^{-1}|$ is

1. 1
2. 2
3. 4
4. 8

Q.

What is the adjoint of the matrix $\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 1\\ 2& 3& 4\end{array}\right]$?

1. $\left[\begin{array}{ccc}1& -2& 1\\ 1& -2& 1\\ -1& 2& -1\end{array}\right]$

2. $\left[\begin{array}{ccc}1& 1& -1\\ -2& -2& 2\\ 1& 1& -1\end{array}\right]$

3. $\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& 3\\ 3& 1& 4\end{array}\right]$

4. $\left[\begin{array}{ccc}-2& 1& -1\\ -2& 1& 1\\ 2& 1& -1\end{array}\right]$

Q. If A = $\left[\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right]$, then det (Adj (Adj A)) is

1. $(14{)}^4$
2. $(14{)}^6$
3. $(14{)}^9$
4. $(14{)}^2$

Q. If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right],\text{then}adj(3A^2+12A)$ is equal to

1. $\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$
2. $\left[\begin{array}{cc}51& 84\\ 63& 72\end{array}\right]$
3. $\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$
4. $\left[\begin{array}{cc}72& -84\\ -63& 51\end{array}\right]$

Q. If $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and A adj $A=A{A}^T$, then 5a + b is equal to

1. 5
2. 4
3. 13
4. -1

Q.

If A is a square matrix of order 4 and the value of |A| is equal to 2. Then the value of |Adj(A)| is,

1. 8

2. 16

3. 2

4. $\sqrt{2}$

Q. Match the properties of adjoint of a matrix :-

1. ${\left(AdjA\right)}^T$
2. $AdjB.AdjA$
3. ${\left(A\right)}^{{}^{N-1}}$ , N is order of $\left(A\right)$