Equality of Matrices

Q.

Whats the trace of a Matrix?

Q.

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

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

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

3. $p=m,q=n$

4. None of these

Q. Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\text{and}B=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],a,b,ϵN.$ Then

1. there exists infinitely many B's such that AB = BA.
2. there cannot exists any B such that AB = BA.
3. there exists more than one but finite number of B's such that AB = BA.
4. there exists exactly one B such that AB = BA.

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

Q.

If $A=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right],$ Then the value of $\alpha$ for which $A^2=B$ is

1. 1

2. -1

3. $i$

4. no real values of $\alpha$

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. 4

2. none of these

3. 2

4. 1

Q.

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

1. $AB$

2. $2AB$

3. $2BA$

4. $A+B$

Q.

Matrix A is such that $A^2=2A-I,$ where I is the Identity matrix. Then for $n\ge 2,A^n$ is equal to

1. $nA-(n-1)I$

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

3. $nA-I$

4. $2^{n-1}A-I$

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. If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$, where I is a matrix satisfying the equation $A{A}^T=9I,$ is $3\times 3$ identity matrix, then the ordered pair (a, b) is equal to

1. (-2, 1)
2. (2, 1)
3. (-2, -1)
4. (2, -1)

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. If $O$ is a point within $\Delta$ABC then show that :
1) $AB+AC=OB+OC$
2) $AB+BC+CA>OA+OB+OC$
3) $OA+OB+OC>\frac{1}{2}(AB+BC+CA)$.

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. 12
2. 7
3. 6
4. 0

Q. If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying the equation $A{A}^T=9I$, where $I$ is $3\times 3$ identity matrix, then the ordered pair $(a,b)$ is equal :

1. $(2,-11)$
2. $(-2,1)$
3. $(2,1)$
4. $(-2,-1)$

Q. If $\left|\begin{array}{ccc}x+1& x+2& x+3\\ x+2& x+3& x+4\\ x+a& x+b& x+c\end{array}\right|=0,$ then a, b, c are in

1. None of these
2. A.P.
3. G.P.
4. H.P.

Q. Show that the matrix $A=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$ satisfies the equation $A^3-4A^2+A=0$.

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 ___

Q. If $f(a+b-x)=f\left(x\right),$ then ${\int }_a^{b}xf\left(x\right)dx$ is equal to :

Q. If the product of n matrices $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]......\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$ is equal to the matrix $\left[\begin{array}{cc}1& 378\\ 0& 1\end{array}\right]$ then the value of $n$ is equal to

1. $26$
2. $377$
3. $27$
4. $378$

Q.

AB + BC + CD > DA

If the above statement is true, then enter 1 or else enter 0.