# Linear Algebra Questions and Answers – Real Matrices: Symmetric, Skew-symmetric, Orthogonal Quadratic Form

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This set of Linear Algebra and Vector Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Real Matrices: Symmetric, Skew-symmetric, Orthogonal Quadratic Form”.

1. Who introduced the term matrix?
a) James Sylvester
b) Arthur Cayley
c) Girolamo Cardano
d) Paul Erdos

Explanation: The term ‘matrix’ was introduced by James Sylvester during the 19th century. Later in 1850, Arthur Cayley developed the algebraic aspect of matrices in two of his papers.

2. Which among the following is listed under the law of transposes?
a) (AT)T = AI
b) (A*)* = A*
c) (cA)T = cAT
d) (AB)T = ATBT

Explanation: The following are listed under the law of transposes:
(A*)* = A
(AT)T = A
(AB)T = BT AT
(cA)T = cAT
(A ± B)T = AT ± BT (and for ∗)
If A is symmetric, A = AT

3. The matrix which remains unchanged under transposition is known as skew symmetric matrix.
a) False
b) True

Explanation: The matrix which remains unchanged under transposition is known as symmetric matrix.
For example, if we consider a symmetric matrix A = $$\begin{bmatrix} 2 & 3 & 4 \\ 3 & 5 & 6\\ 4 & 6 & 7 \end{bmatrix}$$ and take the transpose of it, we get,
AT = $$\begin{bmatrix} 2 & 3 & 4 \\ 3 & 5 & 6 \\ 4 & 6 & 7 \end{bmatrix}$$

4. Find the values of x and y in the matrix below if the matrix is a skew symmetric matrix.
P = $$\begin{bmatrix} 0 & y & -4\\ -5 & 0 & 8 \\ x+y&-8 & 0 \end{bmatrix}$$
a) x = -1, y = 5
b) x = -9, y = -5
c) x = 9, y = 5
d) x = 1, y = -5

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,
y = 5,
$$x+y = 4 \rightarrow x+5=4 \rightarrow x = -1$$

5. Which of the following is known as the reversal rule?
a) (AB)-1 = B-1A-1
b) (AB)-1 = A-1 B-1
c) (BA)-1 = B-1 A-1
d) (BA)-1 = B-1 A

Explanation: The reversal rule of matrix multiplication states that, ‘the inverse of the product of two matrices is equal to the product of their individual inverses, taken in the reverse order’.

6. Every Identity matrix is an orthogonal matrix.
a) True
b) False

Explanation: An orthogonal matrix can be defined as ‘a matrix having its entries as orthogonal unit vectors’ or it can also be defined as ‘a matrix whose transpose is equal to its inverse’. Since this property is satisfied by an identity matrix, every identity matrix is an orthogonal matrix.

7. Which of the following matrix is 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) $$\begin{bmatrix} 0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33 \\ 0.67 & 0.33 & 0.67 \end{bmatrix}$$
c) $$\begin{bmatrix} 0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33 \\ 0.67 & 0.33 & 0.67 \end{bmatrix}$$
d) $$\begin{bmatrix} 0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33 \\ 0.67 & 0.33 & 0.67 \end{bmatrix}$$

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
$$\begin{bmatrix} 0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33 \\ 0.67 & 0.33 & 0.67 \end{bmatrix}$$
$$\begin{bmatrix} 0.33 & -0.67 & 0.67 \\ 0.67 & 0.67 & 0.33\\ -0.67 & 0.33 & 0.67 \end{bmatrix}$$ = $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

8. Find the symmetric matrix for the given quadratic form, $$Q = 2x_1^2 + 2x_2^2 + 5x_3^2 – 3x_1 x_2+7x_3 x_1.$$

a) $$\begin{bmatrix} 1 & \frac{-7}{2} & \frac{3}{2} \\ \frac{-7}{2} & 2 & 0 \\ \frac{3}{2} & 0 & \frac{5}{2}\end{bmatrix}$$
b) $$\begin{bmatrix} 1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix}$$
c) $$\begin{bmatrix} 1 & \frac{3}{2} & \frac{-7}{2}\\ \frac{3}{2} & 2 & 0 \\ -\frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix}$$
d) $$\begin{bmatrix} 1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & 5\end{bmatrix}$$

Explanation: The following steps need to be followed to obtain the symmetric matrix:
Step 1: There are three variables in the given quadratic equation. Hence, the symmetric matrix to be formed should be of dimension 3×3 and the general form can be written as,
Q = $$\begin{bmatrix}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix}$$
Step 2: Place the square term coefficients of the quadratic equation (2, 2, 5) on the diagonal of the matrix.
Q = $$\begin{bmatrix}2 & c_{12} & c_{13} \\ c_{21} & 2 & c_{23} \\ c_{31} & c_{32} & 5\end{bmatrix}$$
Step 3: Place the remaining coefficients of $$x_i x_j \,at\, c_{ij},$$ i.e. coefficient of x1 x2 (-3) at c12 and so on.
Q = $$\begin{bmatrix}2&-3 & 0 \\ 0 & 2 & 0 \\ 7 & 0 & 5\end{bmatrix}$$
Step 4: For a symmetric matrix, S = $$\frac{1}{2} (Q+ Q^T)$$
S = $$\frac{1}{2} \begin{bmatrix}2&-3 & 0 \\ 0 & 2 & 0 \\ 7 & 0 & 5\end{bmatrix} + \begin{bmatrix}2 & 0 & 7 \\ -3 & 2 & 0 \\ 0 & 0 & 5\end{bmatrix}$$
$$S = \frac{1}{2} \begin{bmatrix} 2 & -3 & 7 \\ -3 & 4 & 0 \\ 7 & 0 & 10\end{bmatrix}$$
S = $$\begin{bmatrix}1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix}$$

9. Which of the following is known as Hadamard matrix?
a) $$\begin{bmatrix}1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\end{bmatrix}$$
b) $$\begin{bmatrix}1 & -1 & 1 & -1\\ 1 & -1 & 1 & -1\\ 1 & -1 & 1 & -1\\ 1 & -1 & 1 & -1 \end{bmatrix}$$
c) $$\begin{bmatrix}1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \end{bmatrix}$$
d) $$\begin{bmatrix}0 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}$$

Explanation: Hadamard matrix is named after a famous French mathematician, Jacques Hadamard. It is defined as ‘a square matrix whose entries are only 1 or -1 and whose column (or row) vectors orthogonal’.

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

Explanation: To prove the above statement, let us consider an example,
A = $$\begin{bmatrix} 0 & -4 & 1 \\ 4 & 0 & -3 \\-1 & 3 & 0\end{bmatrix}$$

Therefore, A + A = $$\begin{bmatrix} 0 & -4 & 1 \\ 4 & 0 & -3 \\ -1 & 3 & 0\end{bmatrix}$$ + $$\begin{bmatrix}0&-4 & 1 \\ 4 & 0&-3 \\-1 & 3 & 0\end{bmatrix}$$ = $$\begin{bmatrix} 0 & -8 & 2 \\ 8 & 0 & -9\\ -2 & 6 & 0\end{bmatrix}$$ which is also a skew-symmetric matrix.

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