Solve your IGNOU Doubts

Question:

Find the orthogonal canonical reduction of the quadratic form x² - 2y² + z² + 2xy + 6yz and its principal axes. Also, find the rank and signature of the quadratic form.

See Answer →

Question:

Let (x₁, x₂, x₃) and (y₁, y₂, y₃) represent the coordinates with respect to the bases

$B_1 = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ , $B_2 = \{(1, 0, 0), (0, 1, 2), (0, 2, 1)\}$ . If

$Q(X) = x_1^2 + 2x_1x_2 + 2x_2x_3 + x_2^2 + x_3^2$ , find the representation of Q in terms of (y₁, y₂, y₃).

See Answer →

Question:

) Are there values of $a \in \mathbf{C}$ for which the matrix

$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 1/\sqrt{2} & a \end{bmatrix}$

See Answer →

Question:

Find the values of $a, b \in \mathbf{C}$ for which the matrix

$$\begin{bmatrix} 1 & i & 1+i \\ a & 0 & b \\ 1-i & 2+i & 1 \end{bmatrix}$

is Hermitian.

See Answer →

Question:

) Using the fact that $\text{det}(AB) = \text{det}(A) \text{det}(B)$ for any two matrices A and B, prove the identity

$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$

See Answer →

Question:

Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.

See Answer →

Question:

For the following matrices, check whether there exists an invertible matrix P such that P^-1AP is diagonal. When such a P exists, find P.

i) $A = \begin{bmatrix} 0 & 1 & -3 \\ 2 & -1 & 6 \\ 1 & -1 & 4 \end{bmatrix} \quad \text{ii) } B = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}$

See Answer →

Question:

Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.

$$\begin{aligned} 2a + 3b + 4c + d &= 8 \\ a + 2b + 2c + 2d &= 3 \\ a - b + c + 3d &= 3 \end{aligned}$

See Answer →

Question:

Show that, if A is any $n \times n$ matrix with real entries, then there is a $n \times n$ symmetric matrix S and a $n \times n$ skew symmetric matrix S' such that $A = S + S'$ .

See Answer →

Question:

Is the matrix of the linear operator T non-singular? Justify your answer.

See Answer →

Question:

Write down the matrix of T on W w.r.t the basis S.

See Answer →

Question:

Check that $T(W) \subset W$ .

See Answer →

Question:

Let $W = [S]$ and let $T: V \to V$ be the function defined by

$\quad \quad T(f(x)) = \frac{d^2}{dx^2}(f(x)) + 2\frac{d}{dx}(f(x))$ .
$\quad$

See Answer →

Question:

Check that S is a linearly independent set over $\mathbf{R}$ . (Hint: Consider the equation

$\quad \quad a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x = 0$ .
$\quad$ Put $x = 0, \pi, \frac{\pi}{2}, \frac{\pi}{4}$ , etc. and solve for a_i. )

See Answer →

Question:

Let V be the set of all functions that are twice differentiable in $\mathbf{R}$ and

$\quad S = \{\cos x, \sin x, x\cos x, x\sin x\}$ .

See Answer →

Question:

Find the direction cosines of the perpendicular from the origin to the plane

$\quad \mathbf{r} \cdot (6\mathbf{i} + 4\mathbf{j} + 2\sqrt{3}\mathbf{k}) + 2 = 0$

See Answer →

Question:

Find the direction cosines of the perpendicular from the origin to the plane

See Answer →

Question:

Check that $(f \circ g)(x) = x$ for $x \in \mathbf{R} \setminus \{2\}$ and $(g \circ f)(x) = x$ for $x \in \mathbf{R} \setminus \{-1\}$ .

See Answer →

Question:

Check that $g: \mathbf{R} \setminus \{2\} \to \mathbf{R}$ given by $g(x) = \frac{x-1}{2-x}$ is well defined and 1 - 1.
$\quad \quad$ Further, check that $g(x) \neq -1$ for any $x \in \mathbf{R}$ .