Equality of Matrices

Q.

If matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ where a, b, c are real positive numbers, abc = 1 and $A^{T}A=I,$ then the value of $a^3+b^3+c^3$ is ___

Q. Let $a,b\text{and}c$ be three real numbers satisfying
$\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{ccc}1& 9& 7\\ 8& 2& 7\\ 7& 3& 7\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ . . . (i)

Let $\omega$ be a solution of $x^3$ - 1 = 0 with Im $\omega$ > 0. If a = 2 with b and c satisfying Eq. (i) then the value of $\frac{3}{{\omega }^a}+\frac{1}{{\omega }^b}+\frac{3}{{\omega }^c}$ is

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

Q.

$LetA=\left[\begin{array}{cc}0& \alpha \\ 0& 0\end{array}\right]and(A+I{)}^{50}-50A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],Then,thevalueofa+b+c+dis$

1. 2

2. 1

3. 4

4. none of these

Q. Let $a,b\text{and}c$ be three real numbers satisfying
$\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{ccc}1& 9& 7\\ 8& 2& 7\\ 7& 3& 7\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ . . . (i)

Let b = 6, with a and c satisfying Eq. (i). If $\alpha and\beta$ are the roots of the quadratic equation $a{x}^2$ + bx + c = 0, then ${\sum }_{n=0}^{\infty }{(\frac{1}{\alpha }+\frac{1}{\beta })}^n$ is equal to

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

Q. If $h\left(x\right)=5-9x$ and $h^{-1}\left(x\right)$ is obtained in the form $\frac{b-ax}{9}$, then the quadratic equation with roots $a$ and $b$ is

1. $x^2-6x+5=0$
2. $x^2+6x+5=0$
3. $x^2-6x-7=0$
4. $x^2-7x+10=0$

Q. Let $a,b\text{and}c$ be three real numbers satisfying
$\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{ccc}1& 9& 7\\ 8& 2& 7\\ 7& 3& 7\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ . . . (i)


If the point P(a, b, c), with reference to Eq. (i), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

1. \N
2. 12
3. 7
4. 6

Q. Let $A$ and $B$ be two $n\times n$ matrices such that $det\left(A\right)\ne 0,A+B=(AB{)}^2$ and $BAB=A+I$, then
$\left(I\right)A^{-1}=(A^4-I)\left(II\right)B^5-A^5=A+B\left(III\right)A^9=A^4+A+I\left(IV\right)A^2{B}^2=B{A}^{2}B$

Which of the following is correct?
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. Only $\left(I\right)$ is true
2. Only $\left(I\right)$ and $\left(IV\right)$ are true
3. Only $\left(I\right),\left(III\right)$ and $\left(IV\right)$ are true
4. All are true

Q. If A = $\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]andB=\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$, then value of $\alpha$ for which $A^2$ = B, is

1. 1
2. -1
3. 4
4. non real values

Q.

Two matrices $A=[a_{ij}{]}_{p\times q},B=[b_{ij}{]}_{m\times n}$ are equal if.

1. p=n, q=m and $a_{ij}=b_{ij}\forall i,j$

2. $p=m,q=n$

3. p=m, q=n and $a_{ij}=b_{ij}\forall i,j$

4. None of these

Q. Let M be a $3\times 3$ matrix satisying $M\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right],M\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ -1\end{array}\right],andM\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 12\end{array}\right].$
Then, the sum of the diagonal entries of M is ___