Adjoint of a Matrix

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. 8
4. 4

Q. Let matrix $A=\left[\begin{array}{ccc}1& 1& m\\ p& a& n\\ q& r& 1\end{array}\right],$ $a,m,n,p,q,r,\in R$ be an idempotent matrix. Then the value of $|A|+\frac{|A|}{tr\left(A\right)}+\frac{|A{|}^2}{\left(tr\right(A^2){)}^2}+\frac{|A^3|}{\left(tr\right(A^3){)}^3}+\cdots +\infty \text{terms,}$ $($where $|A|\ne 0$ and $tr\left(A\right)\ne 0)$ is equal to

1. $\frac{3}{2}$
2. $\frac{4}{3}$
3. $2$
4. $1$

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

1. 2

2. $\sqrt{2}$

3. 8

4. 16

Q.

If P = $\left(\begin{array}{ccc}1& \propto & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right)$ is the adjoint of a 3x3 matrix A and det(A)=4, then α is equal to

1. 4

2. 11

3. 5

4. 0

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

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

1. $\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$
2. $\left[\begin{array}{cc}51& 84\\ 63& 72\end{array}\right]$
3. $\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$
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. 13
2. -1
3. 5
4. 4

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 $\left(A\right)$

Q. Find the minors and cofactors of elements$a_{23},a_{32}\mathrm{and}{a}_{13}$of matrix $A=\left(a_{\mathrm{ij}}\right]=\left(\begin{array}{ccc}5& 6& 7\\ 5& 2& 3\\ 4& 8& 9\end{array}\right].$

1. $M_{23},M_{32,}{M}_{13}=16,-20,32C_{23},C_{32,}{C}_{13}=-16,20,32$
2. $M_{23},M_{32,}{M}_{13}=16,-20,32C_{23},C_{32,}{C}_{13}=-16,20,-32$
3. $M_{23},M_{32,}{M}_{13}=16,-20,-32C_{23},C_{32,}{C}_{13}=-16,20,32$
4. $M_{23},M_{32,}{M}_{13}=16,-20,32C_{23},C_{32,}{C}_{13}=-16,-20,32$