Involutory Matrix

Q. Let $A$ be a square matrix of order $3$ whose all entries are $1$ and let $I_3$ be the identity matrix of order $3$. Then the matrix $A-3I_3$ is

1. invertible
2. orthogonal
3. non-invertible
4. real skew symmetric matrix

Q. Let $A$ be a $2\times 2$ matrix with real entries, Let I be the $2\times 2$ identity matrix. Denote by $tr\left(A\right),$ the sum of diagonal entries of $A$. Assume that $A^2=I$
Statement $1:$ If $A\ne I$ and $A\ne -I$, then det$A=-1$
Statement $2:$ If $A\ne I$ and $A\ne -I$, then $tr\left(A\right)\ne 0,$ then which of the following is correct

1. Statment $1$ is false, statement $2$ is true.
2. Both statement $1$ & $2$ are true and statement $2$ is a correct explanation for statement $1$
3. Both statement $1$ & $2$ are true but statement $2$ is not a correct explanation for statement $1$.
4. Statement $1$ is true, statement $2$ is false

Q. If A is involutory matrix and I is unit matrix of same order, then (I - A)(I + A) is

1. Zero matrix
2. A
3. I
4. 2A

Q. Let $A=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}{9}^2& -{10}^2& {11}^2\\ {12}^2& {13}^2& -{14}^2\\ -{15}^2& {16}^2& {17}^2\end{array}\right]$, then the value of $A^{\prime }BA$ is

Q. Let $tr\left(X\right)$ and $adj\left(X\right)$ denote the trace and adjoint of a square matrix $X.$ If $M$ is a non-singular square matrix of order $3$ such that $M^{-1}=\left[\begin{array}{ccc}3& 4& 5\\ 4& 5& 3\\ 5& 3& 4\end{array}\right],$ then the value of $|15tr\left(adj\right(M\left)\right)|$ is

Q. The trace of a square matrix is defined as the sum of the principal diagonal elements. For real numbers $a$ and $b,$ if the trace of matrices $A=\left[\begin{array}{cc}2a^2& 5\\ 3& 9-6b\end{array}\right]$ and $B=\left[\begin{array}{cc}-b^2& 2\\ 3& 8a-8\end{array}\right]$ are equal, then $2a-b$ is equal to

1. $0$
2. $1$
3. $2$
4. $4$

Q. If $P$ is a non null matrix of order $3$ with real number as entries, such that $P^3=0,$ where $O$ is null matrix of order $3$ then (where $I$ is the identity matrix of order $3$)

1. det. of $(4P^2-2P+I)$ is a non zero number.
2. $I-2P$ is an invertible matrix
3. If matrix $P$ has all integer entries, then $|I-4P^2|=0$
4. If matrix $P$ has all integer entries, the absolute value of $|I-4P^2|=1$

Q. If $A=\left[\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right]$, then $A^{100}=$

1. $$\left[\begin{array}{ccc}0& 0& x^{100}\\ 0& x^{100}& 0\\ x^{100}& 0& 0\end{array}\right]$$
2. $$\left[\begin{array}{ccc}{x}^{100}& 0& 0\\ 0& x^{100}& 0\\ 0& 0& x^{100}\end{array}\right]$$
3. $$\left[\begin{array}{ccc}0& x^{100}& 0\\ 0& 0& x^{100}\\ x^{100}& 0& 0\end{array}\right]$$
4. $$\left[\begin{array}{ccc}0& x^{100}& 0\\ x^{100}& 0& 0\\ 0& 0& x^{100}\end{array}\right]$$

Q. If $A$ is a $3$rd order square diagonal matrix of non-positive entries such that $A^2=I$, then

1. $A^{-1}$ does not exist
2. Value of $|A|$ is $-1$
3. There may exist some diagonal elements such that Tr$\left(A\right)$ is zero
4. $|3\text{Adj}\left(2A\right)|=1728$

Q. If $A$ is a $3$rd order square diagonal matrix of non-positive entries such that $A^2=I$, then

1. $A^{-1}$ does not exist
2. $|3\text{Adj}\left(2A\right)|=1728$
3. Value of $|A|$ is $-1$
4. There may exist some diagonal elements such that Tr$\left(A\right)$ is zero

Q.

If A is an idempotent matrix and A + B =I, then which of the following is true?

1. B is idempotent

2. $AB=O$

3. $BA=O$

4. all of these

Q. If $A$ is an idempotent matrix of index $2,$ then which of the following statement(s) is/are true ?
$I.$ $I+A$ is idempotent.
$II.$ $I-A$ is idempotent.

1. Only $I$
2. Only $II$
3. both $I$ and $II$
4. neither $I$ nor $II$

Q.

$IfA=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right](wherebc\ne 0)satisfiestheequation{x}^2+k=0,then$

1. $k=|A|$

2. $k=-|A|$

3. $ad=bc$

4. $a-d=0$

Q. Which of the following matrix$(\ne I)$ may not invertible :

1. idempotent
2. orthogonal
3. involutory
4. None of the above

Q.

If A is involutary then$\left|A\right|=1or-1$

1. True

2. False

Q.

If A is involutary then$\left|A\right|=1or-1$

1. False

2. True

Q. If $A$ is an idempotent matrix and $(2I+A{)}^4=aI+bA,(I\text{is unit matrix}),a,b\in R.$ Then $(a+b)=$

Q. The region represented by $z=x+iy\in C:\left|\begin{array}{c}z\end{array}\right|-Re\left(z\right)\le 1$ is also given by the inequality:

1. $y^2\le 2(x+\frac{1}{2})$
2. $y^2\le x+\frac{1}{2}$
3. $y^2\le 2(x+1)$
4. $y^2\le x+1$

Q.

If A is an idempotent matrix and A + B =I, then which of the following is true?

1. B is idempotent

2. $AB=O$

3. $BA=O$

4. all of these

Q.

$\text{A =}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\text{is an involutary matrix.}$

1. True

2. False

Q. Let $A$ be a square matrix all of whose entries are integers, then which of the following is true?

1. If $\left|A\right|\ne \pm 1$, then $A^{-1}$ exist & all its entries are non-integer
2. If $\left|A\right|=\pm 1$, then $A^{-1}$ exist & all its entries are integer
3. If $\left|A\right|=\pm 1$, then $A^{-1}$ need not exist
4. If $\left|A\right|=\pm 1$, then $A^{-1}$ exist but all its entries are not necessarily integers.

Q. If $A$ and $B$ are any two matrices such that $AB=0$ and $A$ is non-singular, then

1. $B=0$
2. $B$ is singular
3. $B$ is non-singular
4. $B=A$

Q. If A is a non-diagonal involutory matrix, then

1. A - I = 0
2. A + I = 0
3. A - I is non zero singular
4. none of these

Q. If $A$ and $B$ are non singular matrix of same order such that $|AB|=10,|A|=5.$ Then the value of $(|A|-|B|{)}^2$ is:

1. $9$
2. $49$
3. $41$
4. $4$

Q. If A is a non-diagonal involutory matrix, then

1. A + I = 0
2. A - I is non zero singular
3. none of these
4. A - I = 0

Q.

$IfA\times Ai.e.,A^2=IThenAissaidtobeinvolutarymatrix$

1. False

2. True

Q.

If A is an idempotent matrix and A + B =I, then which of the following is true?

1. all of these

2. B is idempotent

3. $AB=O$

4. $BA=O$

Q.

If A is involutary then$\left|A\right|=1or-1$

1. True

2. False

Q. Statement-1 : If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, then $adj\left(adjA\right)=A$
Statement -2 : $|adj\left(adjA\right)|=|A{|}^{(n-1{)}^2}$, where A is a n-rowed non-singular square matrix

1. Statement -1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1
2. Statement -1 is True, Statement-2 is True ; Statement-2 is not a correct explanation for Statement-1
3. Statement-1 is True, Statement-2 is False
4. Statement-1 is False, Statement-2 is True

Q.

$\text{A =}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\text{is an involutary matrix.}$

1. True

2. False