Matrices A and B are invertible, with A⁻¹ = [2 1; 1 1] and B⁻¹ = [1 3; 0 1]. What is the entry in row 1, column 2 of (AB)⁻¹?

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Product, then undo

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Matrices $A$ and $B$ are invertible, with $A^{- 1} = [2111]$ and $B^{- 1} = [1031]$ . What is the entry in row $1$ , column $2$ of $(A B)^{- 1}$ ?

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Hint

In the product $A B$ the matrix $B$ acts on a vector first; think about which factor must be undone first.

Solution

The inverse of a product reverses the order: $(A B)^{- 1} = B^{- 1} A^{- 1}$ , since $A B \times B^{- 1} A^{- 1} = I$ .
The wanted entry is row $1$ of $B^{- 1}$ , which is $(1, 3)$ , times column $2$ of $A^{- 1}$ , which is $(1, 1)$ .
Their dot product is $1 \times 1 + 3 \times 1 = 4$ .
So the entry in row $1$ , column $2$ of $(A B)^{- 1}$ is $4$ .

Answer: $4$