- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.6. $\quad$ Exercises
10. $\quad$ Calculate det $A,$ det $(A^{-1}),$ and $A^{-1}$ for the following upper triangular matrix: $$A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 2 & 3 \\ 0 & 0 & 0 & 2 \end{bmatrix}.$$
Solution. $\quad$ \begin{align*} \det A &= 2 \cdot 2 \cdot 2 \cdot 2 = 16. \end{align*} To find the inverse of $A,$ we can apply the Gauss-Jordan elimination process to the following augmented matrix: \begin{align*} \begin{bmatrix} \begin{array}{cccc|cccc} 2 & 3 & 4 & 5 & 1 & 0 & 0 & 0 \\ 0 & 2 & 3 & 4 & 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 3 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 1 \end{array} \end{bmatrix} \end{align*} \begin{align*} \begin{bmatrix} \begin{array}{cccc|cccc} 2 & 3 & 4 & 0 & 1 & 0 & 0 & -5/2 \\ 0 & 2 & 3 & 0 & 0 & 1 & 0 & -2 \\ 0 & 0 & 2 & 0 & 0 & 0 & 1 & -3/2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1/2 \end{array} \end{bmatrix} \end{align*} \begin{align*} \begin{bmatrix} \begin{array}{cccc|cccc} 2 & 3 & 0 & 0 & 1 & 0 & -2 & 1/2 \\ 0 & 2 & 0 & 0 & 0 & 1 & -3/2 & 1/4 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1/2 & -3/4 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1/2 \end{array} \end{bmatrix} \end{align*} \begin{align*} \begin{bmatrix} \begin{array}{cccc|cccc} 2 & 0 & 0 & 0 & 1 & -3/2 & 1/4 & 1/8 \\ 0 & 1 & 0 & 0 & 0 & 1/2 & -3/4 & 1/8 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1/2 & -3/4 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1/2 \end{array} \end{bmatrix} \end{align*} \begin{align*} \begin{bmatrix} \begin{array}{cccc|cccc} 1 & 0 & 0 & 0 & 1/2 & -3/4 & 1/8 & 1/16 \\ 0 & 1 & 0 & 0 & 0 & 1/2 & -3/4 & 1/8 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1/2 & -3/4 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1/2 \end{array} \end{bmatrix}. \end{align*} As such, the inverse of $A$ is given by \begin{align*} A^{-1} &= \begin{bmatrix} \begin{array}{cccc} 1/2 & -3/4 & 1/8 & 1/16 \\ 0 & 1/2 & -3/4 & 1/8 \\ 0 & 0 & 1/2 & -3/4 \\ 0 & 0 & 0 & 1/2 \end{array} \end{bmatrix} \end{align*} And its determinant is the product of its diagonal elements: \begin{align*} \det A^{-1} &= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{16}. \quad \blacksquare \end{align*}