Equality of Matrices

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. $a+d=0$

3. $k=|A|$

4. $ad=bc$

Q.

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

1. $2AB$

2. $2BA$

3. $A+B$

4. $AB$

Q.

Define equal matrices. Give an example.

Q.

A person writes letters to six friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that,

(a) At least two of them are in the wrong envelopes?

(b) All the letters are in the wrong envelopes?

Q.

Question 124 (iv)

Evaluate the following, using distributive property.
$68\times (-17)+(-68)\times 3$

Q.

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ are two matrices such that AB = BA and $c\ne 0$, then value of $\frac{a-d}{3b-c}$ is :

1. 0

2. -2

3. -1

4. 2

Q.

If A =$\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2$ =$\left[\begin{array}{cc}x& y\\ y& x\end{array}\right]$ , then

1. x = 2ab, y = a + b

2. x = a² + b² , y = 2ab

3. x = 2ab, y = a – b

4. x = a² – b² , y = ab

Q.

If A=$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then $A^{-1}$ =

1. None of these

2. 

3. 

4. 

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.

Solve for a, b and c; if
(i) $\left[\begin{array}{cc}-4& 5+5\\ 3& 2\end{array}\right]=\left[\begin{array}{cc}b+4& 2\\ 3& c-1\end{array}\right]$

(ii)$\left[\begin{array}{cc}a& a-b\\ b+c& 0\end{array}\right]=\left[\begin{array}{cc}3& -1\\ 0& 0\end{array}\right]$

Q.

Given $\left[\begin{array}{cc}x& y+2\\ 3& z-1\end{array}\right]=\left[\begin{array}{cc}3& 1\\ 3& 2\end{array}\right]$
; find x, y and z.

Q.

If $A=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]$, $kA=\left[\begin{array}{cc}0& 3a\\ 2b& 24\end{array}\right]$, then the values of k, a, b respectively are ______ .

1. -6, -12, -16

2. -6, 4, -9

3. -6, -4, -9

4. -6, 12, 18

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. $nA-I$

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

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

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

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

Q.

If two matrices are equal, their corresponding elements are factors of each other.

1. True

2. False

Q.

If $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]$ and $A^2$ = I then x = ___

1. 1
2. 2
3. 4

4. 0

Q.

If two matrices are equal, their corresponding elements are factors of each other. State whether true or false.

1. True

2. False

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. $i$

3. no real values of $\alpha$

4. 1

Q. If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

1. $\alpha =a^2+b^2,\beta =2ab$
2. $\alpha =a^2+b^2,\beta =ab$
3. $\alpha =a^2+b^2,\beta =a^2-b^2$
4. $\alpha =2ab,\beta =a^2+b^2$

Q.

Find x and y if $\left(\begin{array}{c}x-8\\ y+2\end{array}\right)=\left(\begin{array}{c}2x+3\\ 3y-6\end{array}\right)$

1. x = 11, y = -4

2. x = -11, y = 4

3. x = 4, y = -11
   

4. x = -4, y = 11

Q.

If A = $\left[\begin{array}{cc}2& 10\\ 21& 1\end{array}\right]$, B =$\left[\begin{array}{cc}-2& 0\\ 4& 5\end{array}\right]$ then find 2A + B.

Q. Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ are two matrices such that AB = BA and $c\ne 0$, then value of $\frac{a-d}{3b-c}$ is :

1. -2
2. -1
3. 0
4. 2

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

Q.

Find the value of x, given that $A^2$ = B,

A= $\left[\begin{array}{cc}2& 12\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}4& x\\ 0& 1\end{array}\right]$
___

Q. Given $\left[\begin{array}{cc}x& y+2\\ 3& z-1\end{array}\right]=\left[\begin{array}{cc}3& 1\\ 3& 2\end{array}\right]$; then find the value of x+y +z.

1. 6
2. 4
3. 5
4. 8

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.

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

1. $\alpha =a^2+b^2,\beta =2ab$

2. $\alpha =a^2+b^2,\beta =ab$

3. $\alpha =a^2+b^2,\beta =a^2-b^2$

4. $\alpha =2ab,\beta =a^2+b^2$

Q.

If $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]$ and $A^2$ = I then x = ___

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

Q.

If $A=\left[\begin{array}{cc}0& 2\\ 3& 5\end{array}\right]$, $B=\left[\begin{array}{ccc}3& 4& 5\\ 2& 3& 1\\ 1& 3& 7\end{array}\right]$, then A + B =

1. 

2. 

3. 

4. None