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[All Steps] Cash flow to bank account balance. The T -vector c represents the cash flow for an interest bearing bank account over T time periods. Positive values

Question: Cash flow to bank account balance. The $T$ -vector $c$ represents the cash flow for an interest bearing bank account over $T$ time periods. Positive values of $c$ indicate a deposit, and negative values indicate a withdrawal. The $T$ -vector $b$ denotes the bank account balance in the $T$ periods. We have $b_{1}=c_{1}$ (the initial deposit or withdrawal) and

\[b_{t}=(1+r) b_{t-1}+c_{t}, \quad t=2, \ldots, T\]

where $r>0$ is the (per-period) interest rate. (The first term is the previous balance plus the interest, and the second term is the deposit or withdrawal.)

Find the $T \times T$ matrix $A$ for which $b=A c$. That is, the matrix $A$ maps a cash flow sequence into a bank account balance sequence. Your description must make clear what all entries of $A$ are.

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[All Steps] Cash flow to bank account balance. The T -vector c represents the cash flow for an interest bearing bank account over T time periods. Positive values

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