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
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 ### Your Answer Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0\$ or $$\beta^2-1=0$$

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