Multiplication of Matrices

Q. If $x^4+\frac{1}{x^4}=194,$find $x^3+\frac{1}{x^3},x^2+\frac{1}{x^2}$and $x+\frac{1}{x}$

Q. Let A and B be square matrices of same order satisfying AB = A and BA = B. Then $A^2{B}^2$ equals, (O being the zero matrix of the same order as B)

1. O
2. I
3. A
4. B

Q. If A and B are two square matrices of order 3 $\times$ 3 which satisfy AB = A and BA = B then $(A+B{)}^7$ is

1. $7(A+B)$
2. ${7.1}_{3\times 3}$
3. $64(A+B)$
4. ${1281}_{3\times 3}$

Q. If A is $3\times 4$ matrix and B is a matrix such that A'B and BA' are both defined. Then B is of the type

1. $4\times 3$
2. $3\times 4$
3. $3\times 3$
4. $4\times 4$

Q.

State the conjugate surd of 9+root7.

Q.

Write the following polynomials in the standard form:

$\sqrt{11}x+7-\frac{3}{2}{x}^4+x^3$

Q.

VSA

Q.

$IfF\left(x\right)=\left[\begin{array}{ccc}cosx& -sinx& 0\\ sinx& cosx& 0\\ 0& 0& 1\end{array}\right],thenF\left(x\right).F\left(y\right)=$

1. $F\left(x\right)+F\left(y\right)$

2. $F(x+y)$

3. $F(x-y)$

4. $F\left(xy\right)$

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

1. $Noneofthese$
2. $\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& b^n& 0\\ 0& 0& c^n\end{array}\right]$
3. $\left[\begin{array}{ccc}na& 0& 0\\ 0& nb& 0\\ 0& 0& nc\end{array}\right]$
4. $\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$

Q.

If y = $2^{\frac{1}{{\log }_{x^4}}}$. Then

1. y = x²

2. y = x^1⁄2

3. y = x

4. y = x³

Q. If A and B are two matrices such that AB = B and BA = A, then

1. $(A^6-B^5{)}^3=A-B$
2. $A-Bisidempotent$
3. $A-Bisnilpotent$
4. $(A^5-B^5{)}^3=A^3-B^3$

Q. Suppose a Matrix A satisfies $A^2-5A+7I=0$ If $A^5=aA+bI$, then the values of 2a + b is.

1. 1453
2. 1155
3. -87
4. -105

Q. If the volume of a cuboid is 3x² − 27, then its possible dimensions are

(a) 3, x², − 27x

(b) 3, x − 3, x + 3

(c) 3, x², 27x

(d) 3, 3, 3

Q. If $A={\left(a_{ij}\right)}_{2\times 2}$, where $a_{ij}=i+j$, then $A$ is equal to:

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

Q. Let g(x)=1 + x - [x] and f(x) =$\{\begin{array}{ccc}-1& x<0& \\ 0,& x=0& \\ 1,& x>0& \end{array}$ then for all x, f[g(x)] is equal to:
(IIT 2001)

1. x
2. 1
3. f(x)
4. g(x)

Q.

$IfA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right],thenfornϵN,A^n=$

1. A, if n is odd

2. A, if n is even

3. I, if n is odd

4. I, if n is even

Q. If $A$ is a square matrix of order $3$ such that $A^2+A+4I=0$, where $0$ is the zero matrix and $I$ is the unit matrix of order $3$, then

1. $A$ is singular and $A+I$ is non-singular
2. $A$ is non-singular and $A+I$ is non-singular
3. $A$ is non-singular and $A+I$ is singular
4. $A$ is singular and $A+I$ is singular
5. $A$ is non-singular and $A-I$ is singular

Q. If $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ then $A^{16}$ =

1. $\left[\begin{array}{cc}256& 0\\ 0& 256\end{array}\right]$
2. $\left[\begin{array}{cc}-16& 0\\ 0& -16\end{array}\right]$
3. $\left[\begin{array}{cc}0& 16\\ 16& 0\end{array}\right]$
4. $\left[\begin{array}{cc}0& 256\\ 256& 0\end{array}\right]$

Q. Let M and N be two 3 $\times$ 3 matrices such that MN = NM. Further, if $M\ne N^{2}and{M}^2=N^4$, then

1. determinant of $(M^2+M{N}^2)$ is 0.
2. there is a 3 $\times$ 3 non - zero matrix U such that $(M^2+M{N}^2)$ U is zero matrix.
3. determinant of $(M^2+M{N}^2)\ge 1$
4. for a 3 $\times$ 3 matrix U, if $(M^2+M{N}^2)$Uequals the zero matrix, then U is the zero matrix

Q. If $A=\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]$ and I is the identity matrix of order 2, then (A - 2I)(A - 3I) =

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

Q.

Solve the following

$256-(144÷12)$

Q.

If $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$, then $A^2=$

1. $\left[\begin{array}{cc}cos2\alpha & sin2\alpha \\ sin2\alpha & cos2\alpha \end{array}\right]$

2. $\left[\begin{array}{cc}cos2\alpha & -sin2\alpha \\ sin2\alpha & cos2\alpha \end{array}\right]$

3. $\left[\begin{array}{cc}cos2\alpha & sin2\alpha \\ -sin2\alpha & cos2\alpha \end{array}\right]$

4. $\left[\begin{array}{cc}-cos2\alpha & sin2\alpha \\ -sin2\alpha & -cos2\alpha \end{array}\right]$

Q. If in the multiplication of $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$ and $\left[\begin{array}{cc}c& d\\ -d& c\end{array}\right],$ $A$ and $B$ are the elements of the first row, then the elements of the second row will be

1. $B,A$
2. $A,B$
3. $-B,A$
4. $B,-A$

Q. If a and b are positive integers such that a^3-b^3=61, then the value of ab i

Q. If $a+b+c$ = 9 and $ab+bc+ca=26$, then the value of $a^3+b^3+c^3-3abc$ is:

1. 729
2. 27
3. 495
4. 29

Q. If $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]$, then $(A+B)(A-B)$ is equal to

1. None of these
2. $A^2-B^2$
3. $A^2+B^2$
4. $A^2-B^2+BA+AB$

Q. Let $A$ and $B$ be two matrices different from $I$ such that $AB=BA$ and $A^n-B^n$ is invertible for some positive integer $n$. If $A^n-B^n=A^{n+1}-B^{n+1}=A^{n+2}-B^{n+2}$, then

1. $I-A$ is singular
2. $I-B$ is singular
3. $A+B=AB+I$
4. $(I-A)(I-B)$ is non-singular

Q.

$If\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]\text{is the square root of second order unit matrix,}Then\alpha ,\beta and\gamma \text{should satisfy the relation}$

1. $1-{\alpha }^2+\beta \gamma =0$

2. ${\alpha }^2+\beta \gamma -1=0$

3. $1+{\alpha }^2+\beta \gamma =0$

4. $1-{\alpha }^2-\beta \gamma =0$

Q. $\left|A_{3\times 3}\right|=3,\left|B_{3\times 3}\right|=-1$ and
$\left|C_{2\times 2}\right|=+2$ then $\left|2ABC\right|=$

1. $2^3\left(6\right)$
2. $2^3(-6)$
3. $noneofthese$
4. $2(-6)$

Q. If f is defined in $\left[\begin{array}{c}1,3\end{array}\right]$ by $f\left(x\right)=x^3+b{x}^2+ax$, such that $f\left(1\right)-f\left(3\right)=0and{f}^{\prime }\left(c\right)=0$ where $c=2+\frac{1}{\sqrt{3}}$, then (a, b)=

1. $\left(\begin{array}{c}2-\frac{1}{\sqrt{3}},2+\frac{1}{\sqrt{3}}\end{array}\right)$
2. $(11,-6)$
3. $(6,11)$
4. $(-6,11)$