Question

Arrange the matrices in increasing order of their determinant values.

Answer 1

Answer:

c, a, b

Step-by-step explanation:

Given

See attachment for matrices

(a)

$D = \left[\begin{array}{cc}\cos\theta &\sin\theta\\ -\sin\theta& \cos\theta\\\end{array}\right]$

The determinant of the matrix is:

$|D |= (\cos\theta * \cos\theta - \sin\theta *- \sin\theta)$

$|D | = \cos^2\theta + \sin^2\theta$

Using trigonometry ratio, we have:

$|D | = 1$

(b)

$\left[\begin{array}{ccc}2&0\\0&2\end{array}\right]$

The determinant of the matrix is:

$|D| = 2 * 2 - 0 * 0$

$|D| = 4 - 0$

$|D| = 4$

(c)

$\left[\begin{array}{ccc}0&i\\-i&0\end{array}\right]$

The determinant of the matrix is:

$|D| = 0 * 0 -(-i * i)$

$|D| = 0 +i^2$

$|D| = i^2$

In complex numbers

$i^2 = -1$

So:

$|D| = -1$

So, the order of the determinants is: c, a, b