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.

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.

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

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

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.

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

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.

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.

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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