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
ANSWERED
asked 2020-11-02
Show that B is the inverse of A.
\(A=\begin{bmatrix}1 & -1 \\-1 & 2 \end{bmatrix} , B=\begin{bmatrix}2 & 1 \\1 & 1 \end{bmatrix}\)

Answers (1)

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.
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