Conjugate of a Matrix

Q.

Who is the father of matrices?

Q. If $\left[\begin{array}{cc}xy& 4\\ z+6& x+y\end{array}\right]=\left[\begin{array}{cc}8& w\\ 0& 6\end{array}\right]$, then find the values of x, y, z and w.

Q.

Which of the given values of x and y make the following pair of matrices equal

(A)

(B) Not possible to find

(C)

(D)

Q.

Let A={a1, a2, a3, a4, a5} and B={b1, b2, b3, b4, b5} where ai's and bi's are school going students . Define a relation from A to B by xRy iff y is a true friend of x . If R={(a1, b1), (a2, b2), (a3, b3), (a4, b4), (a5, b5)} . Prove that R is neither one one nor onto

Q. The value of $x$ which satisfies $\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ x-4& 2x-9& 3x-16\\ x-8& 2x-27& 3x-64\end{array}\right|=0$ is:

Q.

Imaginary part of third element of the second row for the Conjugate of $\left[\begin{array}{ccc}3+4i& 4& 2+5i\\ 1+2i& 2+3i& 3+5i\\ 2+7i& 9& 5\end{array}\right]$ is

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Q.

Let * be any binary operation on a set A. An element a' ϵ A is called an inverse of a ϵ A for the binary operation *, if

1. a * a' = a = a' * a

2. a * e = a' = e * a

3. a * a' = e = a'* a

4. e * a' = a = a' * e

Q. If $\left[\begin{array}{cc}a-b& 2a+c\\ 2a-b& 3c+d\end{array}\right]=\left[\begin{array}{cc}-1& 5\\ 0& 13\end{array}\right]$, find the value of b.

Q. If $A=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$ and C_ij is cofactor of a_ij in A, then value of |A| is given by

(a) a₁₁ C₃₁ + a₁₂ C₃₂ + a₁₃ C₃₃
(b) a₁₁ C₁₁ + a₁₂ C₂₁ + a₁₃ C₃₁
(c) a₂₁ C₁₁ + a₂₂ C₁₂ + a₂₃ C₁₃
(d) a₁₁ C₁₁ + a₂₁ C₂₁ + a₁₃ C₃₁

Q. Give an example of a function
(i) which is one-one but not onto
(ii) which is not one-one but onto
(iii) which is neither one-one nor onto

Q.

Given, find the values of x, y, z and

w.

Q.

If (i, j)th element of matrix A is 2+3i , what is the corresponding element in A* or conjugate of A.

1. 3 + 2i

2. 2 - 3i

3. 3 - 2i

4. -2 - 3i

Q. Write whether f : R → R, given by $f\left(x\right)=x+\sqrt{x^2}$, is one-one, many-one, onto or into.

Q. For what values of a and b if A = B, where

$A=\left[\begin{array}{cc}a+4& 3b\\ 8& -6\end{array}\right],B=\left[\begin{array}{cc}2a+2& b^2+2\\ 8& b^2-5b\end{array}\right]$

Disclaimer: There is a misprint in the question, b² − 5b should be written instead of b² − 56.

Q. Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.

Q. All the elements in a matrix A are complex numbers with imaginary parts not equal to zero. If $A^{\ast }$ is the conjugate of the matrix A, $a_{ij}$ is the general element of matrix A, then what is the general element of the matrix, $\frac{A+A^{\ast }}{2}.$

1. $2lm\left(a_{ij}\right)$
2. $lm\left(a_{ij}\right)$
3. $Re\left(a_{ij}\right)$
4. $\frac{Re\left(a_{ij}\right)}{2}$

Q. If $i{j}^{th}$ element of matrix A is $2+3i,$ then the corresponding element in conjugate of A is .

1. $-2-3i$
2. $2-3i$
3. $-2+3i$
4. $i$

Q. Suppose f₁ and f₂ are non-zero one-one functions from R to R. Is $\frac{f_1}{f_2}$ necessarily one-one? Justify your answer. Here, $\frac{f_1}{f_2}:R\to R$ is given by $\left(\frac{f_1}{f_2}\right)\left(x\right)=\frac{f_1\left(x\right)}{f_2\left(x\right)}$ for all $x\in R$.

Q. If $I_3$ is the identity matrix of order $3$, then $(I_3{)}^{-1}$ is

1. $0$
2. $3I_3$
3. Not necessarily exists
4. $I_3$

Q. Show that the function f : R − {3} → R − {2} given by $f\left(x\right)=\frac{x-2}{x-3}$ is a bijection.

Q.

Imaginary part of third element of the second row for the Conjugate of $\left[\begin{array}{ccc}3+4i& 4& 2+5i\\ 1+2i& 2+3i& 3+5i\\ 2+7i& 9& 5\end{array}\right]$ is

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Q. Let be a function defined by

Then,

(a) f is one-one onto
(b) f is one-one into
(c) f is many one onto
(d) f is many one into

Q.

If, then, if the value of α is

A. B.

C. π D.

Q. All the elements in a matrix A are complex numbers with imaginary parts not equal to zero. If $A^{\ast }$ is the conjugate of the matrix A, $a_{ij}$ is the general element of matrix A, then what is the general element of the matrix, $\frac{A+A^{\ast }}{2}.$

1. $2lm\left(a_{ij}\right)$
2. $lm\left(a_{ij}\right)$
3. $Re\left(a_{ij}\right)$
4. $\frac{Re\left(a_{ij}\right)}{2}$

Q. If and C_ij is cofactor of a_ij in A, then value of |A| is given by

(a) a₁₁ C₃₁ + a₁₂ C₃₂ + a₁₃ C₃₃
(b) a₁₁ C₁₁ + a₁₂ C₂₁ + a₁₃ C₃₁
(c) a₂₁ C₁₁ + a₂₂ C₁₂ + a₂₃ C₁₃
(d) a₁₁ C₁₁ + a₂₁ C₂₁ + a₁₃ C₃₁

Q. If $-1<x<0$, then ${\tan }^{-1}x$ equals?

1. $\pi -{\cos }^{-1}\left(\sqrt{1-x^2}\right)$
2. ${\sin }^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)$
3. $-{\cos }^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)$
4. $cose{c}^{-1}x$

Q.

If A is matrix, then conjugate of A is obtained by taking conjugate of the elements of a particular row or a column.

1. True

2. False

Q.

Find x and y, if

Q. $\left(\begin{array}{cc}2i& 9+2i\\ -9+2i& 3i\end{array}\right]\mathrm{is}a\mathrm{skew}\mathrm{hermitian}\mathrm{matrix}.$

1. False
2. True

Q.

Find x and y, if