Why is it time-variant?

The multiplier t makes the delayed output (t-t0)x(t-t0-1)

It has memory because

Output depends on the input 1 s earlier

Short answer

It is linear, time-variant, causal, and with memory (output depends on past input x(t-1)).

Determine whether or not the given system is linear, time invariant, causal and with or without memory: y(t) = t·x(t-1).

y(t) = t·x(t - 1)

Linear? Response to a·x₁ + b·x₂ is t[a·x₁(t-1) + b·x₂(t-1)] = a·y₁ + b·y₂. Linear.
Time-invariant? Delay the input by t₀: y₁(t) = t·x(t - 1 - t₀). Delay the output: y(t - t₀) = (t - t₀)·x(t - t₀ - 1). These differ because of the multiplier t. Time-variant.
Causal? The output at t depends on x(t - 1), the input one second earlier. No future values are used. Causal.
Memory? The output at time t depends on a past input, x(t - 1), so the system has memory (it is dynamic).

Answer: linear, time-variant, causal, with memory.

✓ Answer checked by Claude Opus 5.5 (high effort)