Adjoint of a Matrix

Q. Let $A$ be a $3\times 3$ real matrix. If $det\left(\text{2 Adj(2 Adj(Adj(2A)))}\right)=2^{41},$ then the value of $det\left(A^2\right)$ equals

Q.

If $a,b,c$ are in GP and $x,y$ are the arithmetic means between $a,b$and $b,c$ respectively, then $\frac{a}{x}+\frac{c}{y}$ is equal to

1. $0$

2. $1$

3. $2$

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

Q.

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

$\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$

Q. Consider a matrix $A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If $\frac{det\left(adj\right(adj\left(adj\right(adjA\left)\right)\left)\right)}{(\alpha -\beta {)}^{16}(\beta -\gamma {)}^{16}(\gamma -\alpha {)}^{16}}=2^{32}\times 3^{16}$, then the number of such $3$-tuples $(\alpha ,\beta ,\gamma )$ is

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. 4
3. 8
4. 16

Q.

If $A$ is matrix that $A^2=A+I$, where $I$ is unit matrix then $A^5$ is equal to

1. $5A+I$

2. $5A+2I$

3. $5A+3I$

4. $5A+4I$

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.

Subtract $4\mathrm{a}-7\mathrm{ab}+3\mathrm{b}+12$ from $12\mathrm{a}-9\mathrm{ab}+5\mathrm{b}-3$.

Q. If $A$ be a non-singular matrix of order $2,$ such that $|A+|A|\text{adj}\left(A\right)|=0,$ then which of the following option(s) is/are always correct ?
(where $\text{adj}\left(A\right)$ is the adjoint of matrix $A$ )

1. $|A|=1$
2. the trace of matrix $A$ is $0.$
   (the trace of a square matrix is the sum of elements on the main diagonal)
3. $|A-|A|\text{adj}\left(A\right)|=2$
4. $|A-|A|\text{adj}\left(A\right)|=4$

Q. Let A, B are two non-singular matrices of order 3 with real entries such that $\text{adj}\left(A\right)=3B$ and $\text{adj}\left(B\right)=2A$ then -

1. $\text{adj}\left(A^{2}B\right)+\text{adj}\left(A{B}^2\right)=36(2A+3B)$
2. $B=6A^{-1}$
3. $AB=12I$
4. $\text{adj}\left(A{B}^{-1}\right)=9A^{-2}$

Q. A and B are two 3*3 matrices such that they are inverse of each other then tr.(5AB+6BA+7(AB)^2 +8(BA)^2) is equal to

Q. If $A=\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right],$ then show that $A^3-23A-40I=O$

Q. Let $L=\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]=P+Q,$ where $P$ is a symmetric matrix & $Q$ is a skew-symmetric matrix, then $P$ is equal to

1. $\left[\begin{array}{ccc}3& 5& 6\\ 5& 6& 4\\ 9& 4& 3\end{array}\right]$
2. $\left[\begin{array}{ccc}2& \frac{7}{2}& 3\\ \frac{7}{2}& 1& 2\\ 3& 2& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}6& 5& 4\\ 3& 6& 3\\ 5& 2& 5\end{array}\right]$
4. $\left[\begin{array}{ccc}1& \frac{7}{2}& 3\\ \frac{7}{2}& 2& 1\\ 3& 2& 1\end{array}\right]$

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

1. 4
2. 11
3. 5
4. 7

Q.

If $\mathrm{A}$ is an invertible matrix of order $2$, then $\left|{\mathrm{A}}^{-1}\right|$ is equal to?

1. $\left|\mathrm{A}\right|$

2. $\frac{1}{\left|\mathrm{A}\right|}$

3. $1$

4. $0$

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. the matrix A(I+A)^-1 IS EQUAL TO( IS |A| IS NOT EQUAL TO 0 , |I+A| IS NOT EQUAL TO 0) (1)(A^-1+I)^-1 (2) A^-I (3)(I+A)A (4)(A^-1 +I)^-1A

Q. If $A_0=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ and $B_0\left[\begin{array}{ccc}-4& -3& -3\\ 1& 0& 1\\ 4& 4& 3\end{array}\right]$
$B_n=adj\left(B_{n-1}\right),n\in N$ and $I$ is an identity matrix order 3, then correct satement is/ are

1. Determinant of $(A_0+A_0^2{B}_0^2+A_0^3+A_0^4{B}_0^4+....10terms$) is equal to zero.
2. $B_1+B_2+....B_{49}$ is equal to 49 $B_0$
3. For a variable matrix X, the equation $A_{0}X=B_0$ will have no solution.
4. None of these

Q. The value of determinant
$\left|\begin{array}{ccc}\sqrt{\left(13\right)}+\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{\left(15\right)}+\sqrt{\left(26\right)}& 5& \sqrt{10}\\ 3+\sqrt{\left(65\right)}& \sqrt{\left(15\right)}& 5\end{array}\right|$ is:

1. $5\sqrt{3}(5+\sqrt{6})$
2. $-5(5-\sqrt{6})$
3. $-5\sqrt{3}(5-\sqrt{3})$
4. $-5\sqrt{3}(5-\sqrt{6})$

Q.

If $A_1,A_2$ are two AM’s between two numbers $a$and $b$and $G_1,G_2$ be two GM’s between same two numbers, then $\frac{(A_1+A_2)}{G_1.G_2}=$

1. $(a+b)/ab$

2. $(a+b)/2ab$

3. $2ab/(a+b)$

4. $ab/(a+b)$

Q. Let $A=[a_{ij}{]}_{4\times 4}$ be a matrix such that $a_{ij}=\{\begin{array}{cc}2,& \text{if}i=j\\ 0,& \text{if}i\ne j\end{array}.$
Then the value of $\left\{\frac{det(adj\left(adjA\right))}{7}\right\}$ is
$\left(\right\{.\}$ represents the fractional part function $)$

1. $\frac{1}{7}$
2. $\frac{2}{7}$
3. $\frac{3}{7}$
4. $\frac{6}{7}$

Q.

If $y=\frac{3a{t}^2}{1+t^3}$ and $x=\frac{3at}{1+t^3}$, then $\frac{dy}{dx}$ is equal to

1. $\frac{t\left(2-t^3\right)}{1-2t^3}$

2. $\frac{t\left(2+t^3\right)}{1-2t^3}$

3. $\frac{t\left(2-t^3\right)}{1+2t^3}$

4. $\frac{t\left(2+t^3\right)}{1+2t^3}$

Q. The value of $\left|\begin{array}{ccc}{1}^2& 2^2& 3^2\\ 2^2& 3^2& 4^2\\ 3^2& 4^2& 5^2\end{array}\right|$ is

1. -8
2. 8
3. 400
4. 1

Q. If A, B, C are three square matrices of third order such that $A=\left[\begin{array}{ccc}x& 0& 2\\ 0& y& 0\\ 0& 0& z\end{array}\right]$ & $|B|=2^2{3}^2,|C|=2,x,y,z\in I^{+}$ & $|adj\left(adjABC\right)|=2^{16}{3}^8{7}^4$ then number of distinct possible matrices A is

Q. For the graph of $f\left(x\right)=\ln \left(x^2–4x+5\right),$ which of the following is/are true:

1. It has a horizontal asymptote.
2. It has inflection poins at $(2,\ln 2)$ and $(3,\ln 2)$
3. It has inflection poins at $(1,\ln 2)$ and $(3,\ln 2)$
4. It has minimum at $x=2$

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 non-singular matrix satisfying $AB-BA=A$, then

1. $det(B+I)=det(B-I)$
2. $det(B+I)=I$
3. $det(B+I)=detA\times detB$
4. $detA=detB$

Q. Verify A ( adj A ) = ( adj A ) A = I .

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. 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. $1$