Question

# Show that B is the inverse of A. A=begin{bmatrix}1 & -1 -1 & 2 end{bmatrix} , B=begin{bmatrix}2 & 1 1 & 1 end{bmatrix}

Matrices
Show that B is the inverse of A.
$$A=\begin{bmatrix}1 & -1 \\-1 & 2 \end{bmatrix} , B=\begin{bmatrix}2 & 1 \\1 & 1 \end{bmatrix}$$

2020-11-03
Step 1
Given the matrices
$$A=\begin{bmatrix}1 & -1 \\-1 & 2 \end{bmatrix} , B=\begin{bmatrix}2 & 1 \\1 & 1 \end{bmatrix}$$
Show that B is the inverse of A.
Step 2
To show that two matrices are inverse it must satisfy the following condition
AB=BA=I
where I is the identity matrix
$$AB=\begin{bmatrix}1 & -1 \\-1 & 2 \end{bmatrix}\begin{bmatrix}2 & 1 \\1 & 1 \end{bmatrix}$$
$$AB=\begin{bmatrix}(1\cdot2)+(-1\cdot1) & (1\cdot1)+(-1\cdot1) \\ (-1\cdot2)+(2\cdot1) & (-1\cdot1)+(2\cdot1) \end{bmatrix}$$
$$AB=\begin{bmatrix}2-1 & 1-1 \\1-1 & 2-1 \end{bmatrix}$$
$$AB=\begin{bmatrix}1 & 0 \\0& 1 \end{bmatrix}=I$$
$$BA=\begin{bmatrix}2 & 1 \\1 & 1 \end{bmatrix}\begin{bmatrix}1 & -1 \\-1 & 2 \end{bmatrix}$$
$$BA=\begin{bmatrix}(2\cdot1)+(1\cdot-1) & (2\cdot-1)+(1\cdot2) \\ (1\cdot1)+(1\cdot-1) & (1\cdot-1)+(1\cdot2) \end{bmatrix}$$
$$BA=\begin{bmatrix}2-1 & -2+2 \\1-1 & -1+2 \end{bmatrix}$$
$$BA=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}=I$$
So AB=BA=I
hence B is the inverse of A.