Exercise 3.3

Exercise 3.3

1. (a) Mention singular and nonsingular matrices from the followings.

(b) If $\left|P\right| = 9$ and $\mathbf{P} = \left[ \begin{array}{ll}3 & 3\\ 1 & 4\mathbf{k} \end{array} \right]$ , then find k.

(c) If $\left|T\right| = 3$ and adj $\mathbf{T} = \left[ \begin{array}{ll}5 & x\\ 4 & 2 \end{array} \right]$ , then find x.

2. (a) Find the multiplicative inverses of these matrices if possible.

$R = \left[ \begin{array}{cc}4 & -1\\ -6 & 2 \end{array} \right], \quad S = \left[ \begin{array}{cc}5\\ 3 \end{array} \right], \quad T = \left[ \begin{array}{cc}25\\ 50 \end{array} \right], \quad U = \left[ \begin{array}{cc}x\\ y \end{array} \right], \quad \text{if} x = y.$

(b) If $\mathbf{R} = \left[ \begin{array}{cc}4 & 1\\ -6 & 2 \end{array} \right]$ then verify that $\mathbf{R}\mathbf{R}^{- 1} = \mathbf{R}^{- 1}\mathbf{R} = \mathbf{I}$ .

3. If $\mathbf{Y} = \left[ \begin{array}{cc}1 & 2\\ 2 & 3 \end{array} \right]$ and $\mathbf{Z} = \left[ \begin{array}{cc}2 & 3\\ 1 & 2 \end{array} \right]$ , then verify that $(\mathbf{Y}\mathbf{Z})^{- 1} = \mathbf{Z}^{- 1}\mathbf{Y}^{- 1}$ .

4. What is relation among $\left|A\right|,\left|A^{-1}\right|$