# Eigen Values and Eigen Vectors Questions and Answers – Transformation (Reduction) of Quadratic Form to Canonical Form

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This set of Linear Algebra Multiple Choice Questions & Answers focuses on “Transformation (Reduction) of Quadratic Form to Canonical Form”.

1. What is the canonical form of the matrix A = $$\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$$?
a) x+xy+y2
b) x2+xy
c) x2+y2
d) x2+xy+y2

Explanation: The quadratic form of the given matrix is,
[x y]$$\begin{bmatrix}1 & 0 \\1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = [x + y y]\begin{bmatrix} x \\ y \end{bmatrix}$$ = x2+xy+y2.

2. The solution of the given matrix equation is _____
$$\begin{bmatrix}3 & 0 & 2\\ 6 & 1 & 1\\ 2 & 8 & 91\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\\ x_3 \end{bmatrix} ₌ \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}$$
a) x1 = 1, x2 = 1, x3 = 2
b) x1 = 0, x2 = 0, x3 = 0
c) x1 = 3, x2 = -1, x3 = -1
d) x1 = 0, x2 = -2, x3 = 4

Explanation: Let A = $$\begin{bmatrix}3 & 0 & 2\\ 6 & 1 & 1\\ 2 & 8 & 91\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\\ x_3 \end{bmatrix} ₌ \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}$$
Hence, the given matrix equation can be written in the form,
AX = B
Multiplying both sides by A-1, we get
X = A-1 B
But since B = 0, X = 0 and hence the solution is,
x1 = 0, x2 = 0, x3 = 0.

3. Which one of the following is not a criterion for linearity of an equation?
a) The dependent variable y should be of second order
b) The derivatives of the dependent variable should be of second order
c) Each coefficient does not depend on the independent variable
d) Each coefficient depends only on the independent variable

Explanation: The two criterions for linearity of an equation are: The dependent variable y and its derivatives of first degree. Each coefficient depends only on the independent variable.
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4. Which among the following does not belong to main types of integrals?
a) Indefinite Integral
b) Proper Definite Integral
c) Improper Definite Integral
d) Real Integral

Explanation: There are generally two types of integrals,
1. Definite Integrals: These are further classified as,

• Proper Definite Integrals
• Improper Definite Integrals

2. Indefinite Integrals

5. Which of the following is true for matrices?
a) (AB)-1 = B-1A-1
b) (AT) = A
c) AB = BA
d) A*I = I

Explanation: The correct forms of the other options are:

• (AT)T = A
• AB ≠ BA
• A*I = A

6. Euler’s integral of the first kind, which is a proper integral, is used to define the gamma function.
a) True
b) False

Explanation: Euler’s integral of the second kind, which is an improper function, is used to define gamma function for integer x>0.
$$Γ(x)= ∫_0^∞ t^{x-1}e^{-t}.dt$$

7. Which of the following matrix is not orthogonal?
a) $$\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix}$$
b) $$\begin{bmatrix}cosx &sinx \\-sinx & cosx\end{bmatrix}$$
c) $$\begin{bmatrix}0.33 & -0.67 & 0.67\\0.67 & 0.67 & 0.33\\ -0.67 & 0.33 & 0.67\end{bmatrix}$$
d) $$\begin{bmatrix}cosx & sinx \\-sinx & -cosx \end{bmatrix}$$

Explanation: Out of the given options, $$\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix}$$ satisfies the condition for orthogonality, i.e. AAT = I
$$\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix} \begin{bmatrix}0.33 & -0.67 & 0.67 \\ 0.67 & 0.67 & 0.33\\-0.67 & 0.33 & 0.67\end{bmatrix}= \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
Since $$\begin{bmatrix}0.33 & -0.67 & 0.67\\ 0.67 & 0.67 & 0.33\\ -0.67 & 0.33 & 0.67\end{bmatrix}$$is the transpose of A, it is also orthogonal.
Coming to $$\begin{bmatrix}cosx & sinx\\ -sin x &cosx\end{bmatrix},$$
$$\begin{bmatrix}cosx & sinx\\-sinx & cosx\end{bmatrix} \begin{bmatrix}cosx & -sinx \\ sinx & cosx\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$
The remaining option $$\begin{bmatrix}cosx & sinx\\ -sinx & -cosx\end{bmatrix},$$which does not satisfy the condition for orthogonality.

8. The determinant of the matrix whose eigen values are 7, 1, 9 is given by ________
a) 7
b) 63
c) 9
d) 17

Explanation: The product of the eigen values of a matrix gives the determinant of the matrix,
Therefore, ∆ = 63.

9. Find the values of x and y in the matrix below if the matrix is a skew symmetric matrix.
P = $$\begin{bmatrix}0 & x+y & 6 \\3 & 0 & 9\\ x & 9 & 0\end{bmatrix}$$
a) x = -6, y = 3
b) x = 3, y = 3
c) x = 6, y = -3
d) x = 0, y = 3

Explanation: The general form of a skew symmetric matrix is given by,
$$\begin{bmatrix}0 & w1 & w2 \\ w1 & 0 & w3 \\ w2 & w3 & 0\end{bmatrix}$$

Therefore, from the given matrix,
x = 6,
x+y = 3 → 6+y=3 → y = -3

10. The sum of two symmetric matrices is also a symmetric matrix.
a) False
b) True

Explanation: To prove the above statement, let us consider an example,
A = $$\begin{bmatrix}1 & 3 & 8\\ 3 & 0 & 5 \\ 8 & 5 & 7 \end{bmatrix}$$

Therefore, A + A =$$\begin{bmatrix}1 & 3 & 8 \\3 & 0 & 5 \\8 & 5 & 7 \end{bmatrix}+ \begin{bmatrix}1 & 3 & 8\\ 3 & 0 & 5\\ 8 & 5 & 7 \end{bmatrix} = \begin{bmatrix}2 & 6 & 16\\ 6 & 0 & 10\\ 16 & 10 & 14 \end{bmatrix}$$ which is also a skew-symmetric matrix.

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