Null Matrix

Q.

If k is a scalar and I is a unit matrix of order 3, then adj(kI) =

1. k³I

2. k²I

3. - k³I

4. - k²I

Q.

Let $\omega$ be a complex cube root of unity with $\omega \ne 0$ and $P=\left[p_{ij}\right]$ be an $n\times n$ matrix with $P_{ij}={\omega }^{i+j}$. Then $P^2=0$ when n is equal to

1. 57

2. 55

3. 58

4. 56

Q.

Given $M=\left(\begin{array}{ccc}9& 7& -8\\ -6& 5& 4\\ -3& 2& -1\end{array}\right)$, Find a matrix N such that M + N is a null matrix.

1. $\left(\begin{array}{ccc}9& 7& -8\\ -6& 5& 4\\ -3& 2& -1\end{array}\right)$

2. $\left(\begin{array}{ccc}-9& -7& 8\\ 6& -5& -4\\ 3& -2& 1\end{array}\right)$

3. $\left(\begin{array}{ccc}-9& -7& -8\\ -6& -5& -4\\ -3& -2& -1\end{array}\right)$

4. $\left(\begin{array}{ccc}9& 7& 8\\ 6& 5& 4\\ 3& 2& 1\end{array}\right)$

Q. If $\omega$ is a cube root of unity, then the value of determinant $\left|\begin{array}{ccc}1+\omega & {\omega }^2& \omega \\ {\omega }^2+\omega & -\omega & {\omega }^2\\ 1+{\omega }^2& \omega & {\omega }^2\end{array}\right|$ is equal to

1. $-1+\omega$
2. ${\omega }^2$
3. $0$
4. $1-\omega$

Q.

Given $A=\left[\begin{array}{cc}-3& 3\\ 2& -2\end{array}\right]\text{and}B=\left[\begin{array}{cc}4& 6\\ 4& 6\end{array}\right]$ Find AB

1. $=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

2. $=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

3. $=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

4. $=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

Q.

Identify the null matrix from the given list

1. $\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]$

2. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

3. $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

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

Q. If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]$, then $A^2$=

1. Unit matrix
2. Null matrix
3. A
4. -A

Q. If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]$, then $A^2$=

1. A
2. -A
3. Unit matrix
4. Null matrix

Q.

Which of the following is a null matrix?

1. 

2. 

3. 

4. 

Q. If $\omega \ne 1$ is a cube root of unity and $x+y+z=a$, $x+\omega y+{\omega }^{2}z=b$, $x+{\omega }^{2}y+\omega z=c$, then

1. $x=(a+b+c)/3$
2. $y=(a+b{\omega }^2+c\omega )/3$
3. $z=(a+b\omega +c{\omega }^2)/3$
4. $x^3+y^3+z^3=a^3/3$

Q.

IF $P=\left[\begin{array}{cc}2& 6\\ 3& 9\end{array}\right]andQ=\left[\begin{array}{cc}3& x\\ y& 2\end{array}\right],$ find x and y such that PQ = null matrix.

Q. Null matrix will have all the elements as zero.

1. False
2. True

Q.

Which of the following statements is correct about a null matrix?

1. It has more than one entries as zero

2. It has only one non-zero entry

3. It has all the entries as zero

4. It is represented as O_mxn where mxn is the order of the matrix

Q. If $A=\left[\begin{array}{cc}1& \omega \\ {\omega }^2& 1\end{array}\right],B=\left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]$ where $\omega$ is complex cube root of unity then $AB+BA+A-2B$:

1. Upper triangular matrix
2. Lower triangular matrix
3. Unit matrix
4. Null matrix

Q. $\left|\begin{array}{ccc}1& 1& 1\\ {}^m{C}_1& {}^{m+1}{C}_1& {}^{m+2}{C}_1\\ {}^m{C}_2& {}^{m+1}{C}_2& {}^{m+2}{C}_2\end{array}\right|=$

1. $m(m+1)$
2. $m(m-1)$
3. $1$
4. $0$