Conjugate of a Matrix

Q.

What is matrix theory?

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

Q.

1. 40^o, 66^o

2. 70^o, 80^o

3. 50^o, 88^o

4. 60^o, 80^o

Q.

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

1. -2 - 3i

2. 3 + 2i

3. 2 - 3i

4. 3 - 2i

Q. If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)⁻¹ = f⁻¹ og ⁻¹.

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. Are the following pairs of sets equal? Give reasons.
(i) A = {2, 3}, B = {x : x is a solution of x² + 5x + 6 = 0};
(ii) A = {x : x is a letter of the word " WOLF"};
B = {x : x is a letter of the word " FOLLOW"}.

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. 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 , what is the corresponding element in A* or conjugate of A.

1. 3 + 2i

2. 2 - 3i

3. 3 - 2i

4. -2 - 3i

Q. If and A = A^T, then
(a) x = 0, y = 5
(b) x + y = 5
(c) x = y
(d) none of these