Answer to Question #223709 in Linear Algebra for Harry

Question

Matrix A is 2x2 matrix

Let A = [(1) (4) (3) (5)]

1.a Find A^-1 by using row operations
1.b Use question 1.a to Express A as a product of elementary matrices

Answer 1

1. a

$$A=\begin{pmatrix} 1& 4\\ 3 & 5 \end{pmatrix}$$

$$\det A=\begin{vmatrix} 1 & 4 \\ 3 & 5 \end{vmatrix}=5-12=-7\not=0=>A^{-1}\ exists$$

$$A^{-1}=\dfrac{1}{-7}\begin{pmatrix} 5 & -4 \\ -3 & 1 \end{pmatrix}$$

$$A^{-1}=\begin{pmatrix} -5/7& 4/7 \\ 3/7 & -1/7 \end{pmatrix}$$

2. Augment the matrix with the identity matrix:

$$\begin{pmatrix} 1 & 4 & & 1 & 0 \\ 3 & 5 & & 0 & 1 \end{pmatrix}$$

$R_2=R_2-3R_1$

$$\begin{pmatrix} 1 & 4 & & 1 & 0 \\ 0 & -7 & & -3 & 1 \end{pmatrix}$$

$R_2=R_2/(-7)$

$$\begin{pmatrix} 1 & 4 & & 1 & 0 \\ 0 & 1 & & 3/7 & -1/7 \end{pmatrix}$$

$R_1=R_1-4R_2$

$$\begin{pmatrix} 1 & 0 & & -5/7 & 4/7 \\ 0 & 1 & & 3/7 & -1/7 \end{pmatrix}$$

$$A^{-1}=\begin{pmatrix} -5/7& 4/7 \\ 3/7 & -1/7 \end{pmatrix}$$

$R_2=R_2+3R_1$

$$\begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$$

$R_2=-7R_2$

$$\begin{pmatrix} 1 & 0 \\ 0 & -7 \end{pmatrix}$$

$R_1=R_1+4R_2$

$$\begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}$$

$$A=\begin{pmatrix} 1& 4\\ 3 & 5 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -7 \end{pmatrix}\begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}$$