Identity Matrix

Q.

Write four solutions for the following equations: $2x+y=7$

Q. Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right].$ Then $A^{2025}-A^{2020}$ is equal to

1. $A^6-A$
2. $A^5$
3. $A^5-A$
4. $A^6$

Q. Let $z=\frac{-1+\sqrt{3}i}{2}$, where $i=\sqrt{-1}$, and r, s $ϵ$ {1, 2, 3}. Let $P=\left[\begin{array}{cc}(-z{)}^r& z^{2s}\\ z^{2s}& z^r\end{array}\right]$ and I be the identity matrix of order 2. Then, the total number of ordered pairs (r, s) for which $P^2$ = -I is ___

Q.

For the matrix, find the numbers a and b such that A² + aA + bI = O.

Q. If $A=\left[\begin{array}{ccc}a& b& c\\ x& y& z\\ p& q& r\end{array}\right],B=\left[\begin{array}{ccc}q& -b& y\\ -p& a& -x\\ r& -c& z\end{array}\right]$ and if $A$ is invertible, then which of the following is/are true:

1. $|adjA|=|adjB|$
2. $|A|=|B|$
3. $|A|=-|B|$
4. $B$ is invertible iff $A$ is invertible

Q.

If $A,B$ are two square matrices of order $n$ and $A$ and $B$ commute $(K$ be a real number), then which of the following is correct

1. $A-KI,B-KI$ commute

2. $A-KI,B-KI$ are equal

3. $A-KI,B-KI$ do not commute

4. both $a$ and $b$ are correct

Q. Let $A$ and $B$ be two square matrices of order $n$ with real entries and $ABA-BAB=I$, Also $A^{2}B+B^{2}A=0$ (where $I$ is the identity matrix), then

1. $A$ and $B$ are invertible
2. $A$ and $B$ can not be invertible
3. either one of $A$ or $B$ is invertible
4. $|A{|}^2|B|=\frac{1}{2^n}$

Q.

1. a scalarmatrix

2. diagonal matrix

3. a zero matrix

4. an identity matrix

Q. The matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 4\end{array}\right]$ is

(a) identity matrix
(b) symmetric matrix
(c) skew-symmetric matrix
(d) diagonal matrix

Q.

If $A,B$ and $C$ are matrices such that $AB=AC,$ then which of the following is true

1. $B=C$ if $A\ne O$

2. $B\ne C$ if $A=O$

3. $B=-C$

4. $B$ need not be equal to $C$

Q.

Which of the following matrix/matrices is/are invertible?

1. A matrix of order 2 x 3

2. An identity matrix

3. A scalar matrix with diagonal elements not equal to 0.

4. A diagonal matrix with diagonal elements as {2, 3, 5, 0, 1}

Q. What is the identity of a neither odd nor even function ?

Q.

Find the inverse of each of the matrices, if it exists.

Q. If A and B are non-singular matrices of same order, then which of the following is/are incorrect?

1. $\left(adj\right(AB){)}^{–1}=(adjB{)}^{–1}(adjA{)}^{–1}$
2. $\left(adjA\right)\left(adj{A}^{–1}\right)=A^2$
3. $(adjA{)}^{–1}=\frac{A}{|A|}$
4. $\left(adjA{)}^{–1}A\right(adjA)=A^{–1}$

Q. If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],$ then $(aI+bE{)}^3=$

1. $aI+bE$
2. $a^{3}I+3a^{2}bE$
3. $a^{3}I+3a{b}^{2}E$
4. $a^{3}I+b^{3}E$

Q. If $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ and $B=A^2+A^4+A^6+A^8+A^{10},$ then the value of $det\left(B\right)$ is equal to

Q. if for a mattix A, A^2+I =0, where I is the identity matrix , then A equal

Q. Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ and I be the identity matrix of order 3. If $Q=\left[q_{ij}\right]$ is a matrix, such that $P^{50}-Q=I$, then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

1. 52
2. 103
3. 201
4. 205

Q. Which of the following is a scalar matrix

1. $\left[\begin{array}{ccc}5& 0& 0\\ 0& 6& 0\\ 0& 0& 7\end{array}\right]$
2. $\left[\begin{array}{ccc}5& 0& 0\\ 0& -5& 0\\ 0& 0& 5\end{array}\right]$
3. $\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& 1\frac{1}{2}& 0\\ 0& 0& 2\frac{1}{2}\end{array}\right]$
4. $\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]$

Q. For a square matrix $A,$ if $A^3=O,$ then $I+A+A^2$ is equal to

1. $I+A$
2. $(I+A{)}^{-1}$
3. $(I-A{)}^{-1}$
4. $I-A$

Q. If $A=\left[\begin{array}{ccc}3& 3& 3\\ 3& 3& 3\\ 3& 3& 3\end{array}\right]$, then $A^3=$

1. $729A$
2. $81A$
3. $243A$
4. $27A$

Q.

(-a-b) whole square.. Identity will open as??

Q. If A is a $3\times 3$ non-singular matrix such that $A{A}^{\prime }=A^{\prime }A$ and $B=A^{-1}{A}^{\prime }$ then BB' equals

1. I + B
2. I
3. $B^{-1}$
4. $\left(B^{-1}\right)$

Q. The values of $K$ for which the matrix

$A=\left[\begin{array}{ccc}1& 0& -K\\ 2& 1& 3\\ K& 0& 1\end{array}\right]$ is invertible, is:

1. All the real values of $K$
2. $K\in [-1,1]$
3. No value of $K$
4. $K\in \{-1,1\}$

Q. The value of $\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|$ is:

1. $2$
2. $-1$
3. $3$
4. $1$

Q. {Q7/30 If }A=(a_{ij}) be a square matrix of order }n and }a_{ij}=(-1)^in_{C_i}x_i}{{}^nC_j, if }n is odd, then trace of }A is equal to

Q.

Which of the following is the correct notation of a null matrix

1. Φ

2. I

3. [Φ]

4. [0]

Q. If $A,B$ are two matrices and $AB=B,BA=A$, then $A^2+B^2$

1. $AB$
2. $A-B$
3. $0$
4. $A+B$

Q. If A is a $3\times 3$ non-singular matrix such that $A{A}^{\prime }=A^{\prime }A$ and $B=A^{-1}{A}^{\prime }$ then BB' equals

1. I + B
2. I
3. $B^{-1}$
4. $\left(B^{-1}\right)$

Q. $\text{Let}P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ and $I$ be the identity matrix of order $3.$ If $Q=\left[q_{ij}\right]$ is a matrix such that $P^{50}-Q=I,$ then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

1. $52$
2. $103$
3. $201$
4. $205$