Adjoint of a Matrix

Q. Matrix $A=\left[\begin{array}{ccc}x& 3& 2\\ 1& y& 4\\ 2& 2& z\end{array}\right]$. If $xyz=60$ and $8x+4y+3z=20$, then $A\left(adjA\right)$ is equal to

1. $\left[\begin{array}{ccc}64& 0& 0\\ 0& 64& 0\\ 0& 0& 64\end{array}\right]$
2. $\left[\begin{array}{ccc}88& 0& 0\\ 0& 88& 0\\ 0& 0& 88\end{array}\right]$
3. $\left[\begin{array}{ccc}68& 0& 0\\ 0& 68& 0\\ 0& 0& 68\end{array}\right]$
4. $\left[\begin{array}{ccc}34& 0& 0\\ 0& 34& 0\\ 0& 0& 34\end{array}\right]$

Q. If a given matrix is symmetric, diagonal or triangular, then its adjoint matrix will also be symmetric, diagonal or triangular respectively.

1. False
2. True

Q.

Find $(-13)+32-8-1$

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. Let $A$ be a $3\times 3$ square matrix. If $B=adj\left(A\right),$ $C=adj\left(adj\right(A\left)\right)$ and $D=adj\left(adj\right(adj\left(A\right)\left)\right),$ then $|adj\left(adj\right(adj\left(adj\right(ABCD\left)\right)\left)\right)|,$ in terms of $|A|$ is

1. $|A^{60}|$
2. $|A^{120}|$
3. $|A^{180}|$
4. $|A^{240}|$

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. Let $P=\left[a_{ij}\right]$ be a $3\times 3$ invertible matrix, where $a_{ij}\in \{0,1\}$ for $1\le i,j\le 3$ and exactly four elements of $P$ are $1$. If $N$ denotes the number of such possible matrices $P$, then which of the following is/are true?

1. Number of divisors of $N$ is even.
2. Sum of divisors of $N$ is $91$.
3. Determinant of $\text{adj}\left(P\right)$ can be $-1$.
4. Determinant of $\text{adj}\left(P\right)$ can be $1$.

Q. Let $A$ be a $3\times 3$ square matrix and matrices $B,C$ and $D$ are related such that $B=adj\left(A\right),$ $C=adj\left(adjA\right)$ and $D=\left(adj\right(adj\left(adjA\right)\left)\right).$ If $det\left(adj\right(adj\left(adj\right(adjABCD\left)\right)\left)\right))=|A{|}^k,$ then the value of $k$ is

Q. Let $A$ be a square matrix of order $3$ whose elements are real numbers and $adj(adj\left(adjA\right))=\left[\begin{array}{ccc}16& 0& -3\\ 0& 4& 0\\ 0& 3& 4\end{array}\right].$ Then the absolute value of $trace\left(A^{-1}\right)$ is

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

Q. If $A$ is a square matrix of order $3$ and $||adj\left(A\right)|\cdot |A|\cdot A|=|A{|}^{\lambda },$ then the value of $\lambda$ is

1. $10$
2. $15$
3. $5$
4. $4$

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

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 $\text{adj}B=A,|P|=|Q|=1,$ then $\text{adj}\left(Q^{-1}B{P}^{-1}\right)$ is

1. $PQ$
2. $QAP$
3. $PAQ$
4. $P{A}^{-1}Q$

Q. If A is a diagonal matrix of order 3 $\times$ 3 is commutative with every square matrix of order 3 $\times$ 3 under multiplication and trace (A) = 12, then

1. $\left|A\right|=64$
2. $\left|A\right|=16$
3. $\left|A\right|=12$
4. $\left|A\right|=0$

Q. For $3\times 3$ matrices $A$ and $B,$ which of the following statements is (are) CORRECT?

1. $AB$ is skew-symmetric if $A$ is symmetric and $B$ is skew-symmetric.
2. $(adjA{)}^T=adj(A^T)$ for all invertible matrix $A.$
3. $AB+BA$ is symmetric for all symmetric matrices $A$ and $B.$
4. $(adjA{)}^{-1}=adj(A^{-1})$ for all invertible matrix $A.$

Q. If A is $6\times 6$ matrix and $||A|adj\left(|A|A\right)|=|A{|}^n$, then $n$ is

1. 40
2. 31
3. 25
4. 41

Q. Let $A$ be a square matrix of order $3$ whose elements are real numbers and $adj(adj\left(adjA\right))=\left[\begin{array}{ccc}16& 0& -3\\ 0& 4& 0\\ 0& 3& 4\end{array}\right].$ Then the absolute value of $trace\left(A^{-1}\right)$ is

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. 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. If $A$ is a square matrix of order $3$ and $||adj\left(A\right)|\cdot |A|\cdot A|=|A{|}^{\lambda },$ then the value of $\lambda$ is

1. $10$
2. $15$
3. $5$
4. $4$

Q. If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to :

1. $4$
2. $11$
3. $5$
4. $0$

Q. If $A$ is a $3\times 3$ matrix such that $|A|=5,$ then the value of $|adj\left(4A\right)|$ is

1. ${16}^3\times 5^2$
2. ${12}^3\times 5^2$
3. ${16}^3\times 6^2$
4. ${14}^3\times 5^2$

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

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

Q. If $A$ is a square matrix such that $A\left(\text{adj}A\right)=\left(\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right),$ then value of $\text{det(adj}A)$ equals to

1. $16$
2. $256$
3. $64$
4. $4$

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