Question

Let T : R
3 → R
3 be defined by
T (x1, x2, x3) = (x1 −x3, x2 −x3, x1).
Is T invertible? If yes, find a rule for T
−1
like the one which defines T.

Answer 1

Answer on Question #81144 – Math – Linear Algebra

Question

Let $T\colon \mathbb{R}^3\to \mathbb{R}^3$ be defined by $T(x_{1},x_{2},x_{3}) = (x_{1} - x_{3},x_{2} - x_{3},x_{1})$.

Is $T$ invertible? If yes, find a rule for $T^{-1}$ like the one which defines $T$.

Solution

If $T\colon \mathbb{R}^3\to \mathbb{R}^3$ is defined by $T(x_{1},x_{2},x_{3}) = (x_{1} - x_{3},x_{2} - x_{3},x_{1})$, we can define

$$T = \left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 0 & 0 \end{array} \right).$$ $$Det(T) = \left| \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 0 & 0 \end{array} \right| = 1 \left| \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array} \right| - 0 - 1 \left| \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right| = 0 + 0 - (0 - 1) = 1 \neq 0$$

Then $T$ is invertible.

Augment the matrix $T$ with identity matrix

$$\left( \begin{array}{ccccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \end{array} \right)$$ $$\left( \begin{array}{ccccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \end{array} \right) \xrightarrow{R_3 - R_1} \left( \begin{array}{ccccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 0 & 1 \end{array} \right)$$ $$\left( \begin{array}{ccccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 0 & 1 \end{array} \right) \xrightarrow{R_1 + R_3} \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & -1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 0 & 1 \end{array} \right)$$ $$\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & -1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 0 & 1 \end{array} \right) \xrightarrow{R_2 + R_3} \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 & 1 & 1 \\ 0 & 0 & 1 & -1 & 0 & 1 \end{array} \right)$$

As can be seen, we have obtained the identity matrix to the left. So, we are done.

$$T^{-1} = \left( \begin{array}{ccc} 0 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & 0 & 1 \end{array} \right)$$

$T^{-1}\colon \mathbb{R}^3\to \mathbb{R}^3$ can be defined by

$$T^{-1}(x_1, x_2, x_3) = (x_3, -x_1 + x_2 + x_3, -x_1 + x_3).$$

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