# Question 4, Exercise 2.1

Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

## Question 4(i)

Find the transpose of the following matrix and identify which one is symmetric and which is skew-symmetric. $$ A=\left[\begin{array}{ccc} 2 & 0 \\ \sqrt{5} & 6 \\ 1 & 9 \end{array}\right]$$

** Solution. **

$$ A^t=\begin{bmatrix} 2 & \sqrt{5} & 1 \\ 0 & 6 & 9 \end{bmatrix}$$

## Question 4(ii)

Find the transpose of the following matrix and identify which one is symmetric and which is skew-symmetric. $$B=\left[\begin{array}{cccc} 1 & 6 & 2 & 0 \end{array}\right] $$

** Solution. **
$$B^t=\left[\begin{array}{c}
1 \\ 6 \\ 2 \\ 0
\end{array}\right] $$

## Question 4(iii)

Find the transpose of the following matrix and identify which one is symmetric and which is skew-symmetric. $$C=\left[\begin{array}{ll} 2 & 6 \\ 9 & 2 \end{array}\right]$$

** Solution. **

$$C^t=\left[\begin{array}{ll} 2 & 9 \\ 6 & 2 \end{array}\right]$$

## Question 4(iv)

Find the transpose of the following matrix and identify which one is symmetric and which is skew-symmetric. $$D=\left[\begin{array}{ccc} 0 & 1 & 9 \\ -1 & 0 & 5 \\ -9 & -5 & 0 \end{array}\right] $$

** Solution. **

$$D^t=\left[\begin{array}{ccc} 0 & -1 & -9 \\ 1 & 0 & -5 \\ 9 & 5 & 0 \end{array}\right] $$

Since $D^t=-D$, therefore $D$ is skew-symmetric.

## Question 4(v)

Find the transpose of the following matrix and identify which one is symmetric and which is skew-symmetric. $$E=\left[\begin{array}{ccc} 3 & -6 & 9 \\ -6 & 2 & 0 \\ 9 & 0 & 0 \end{array}\right] $$

** Solution. **

$$E^t=\left[\begin{array}{ccc} 3 & -6 & 9 \\ -6 & 2 & 0 \\ 9 & 0 & 0 \end{array}\right] $$

Since $E^t=E$, therefore $E$ is symmetric.

## Question 4(vi)

Find the transpose of the following matrix and identify which one is symmetric and which is skew-symmetric. $$F=\left[\begin{array}{ccc} 9 & 0 & 1 \\ 0 & 6 & 3 \\ 0 & 0 & 1 \end{array}\right]$$

** Solution. **

$$F^t=\left[\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 6 & 0 \\ 1 & 3 & 1 \end{array}\right]$$

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