Suppose $A$, $B$, and $X$ are $n \times n$ matrices with $A$, $X$, and $A-AX$ invertible

Suppose $A$, $B$, and $X$ are $n \times n$ matrices with $A$, $X$, and $A-AX$ invertible


J
Asked by 1 year ago
240 points

Suppose $A$, $B$, and $X$ are $n \times n$ matrices with $A$, $X$, and $A-AX$ invertible, and suppose

$(A-AX)^{-1}=X^{-1}B$ (eq. 3)

a. Explain why $B$ is invertible
b. Solve (eq. 3) for $X$. If you need to invert a matrix, explain why that matrix is invertible

Suppose $A
jeffp

1 Answer

Answered by 1 year ago
8.7k points

Explain why $B$ is invertible

$$(A-AX)^{-1}=X^{-1}B \longrightarrow X(A-AX)^{-1} = XX^{-1}B$$ $$ \longrightarrow X(A-AX)^{-1}=B$$

$B$ is the product of two invertible matrices so $B$ must be invertible.

Solve (eq. 3) for $X$. If you need to invert a matrix, explain why that matrix is invertible

$$(A-AX)^{-1}=X^{-1}B \longrightarrow ((A-AX)^{-1})^{-1}=(X^{-1}B)^{-1}$$

$$\longrightarrow A-AX=B^{-1}X \longrightarrow A=AX+B^{-1}X$$

$$\longrightarrow A=(A+B^{-1})X\longrightarrow X=(A+B)^{-1}A$$

where $(A+B)^{-1}$ is invertible since $A$ and $X$ and invertible

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Asked: 1 year ago

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