why do viruses multiply only in the living cells
Virus cannot reproduce without a living cell. Once a virus infects a living cell, it injects its genetic material (DNA or RNA) inside the host cell. This genetic material after entering inside a living cell gets incorporated with the host DNA and starts replicating. The new replicated part of the viral genetic material covers itself with a protein coat and forms new virus. Without a living cell the genetic material of the virus cannot replicate.
Because viruses do not have the 'machinery' to make copies of themselves. At a minimum viruses have a nucleic acid (DNA or RNA) contained within a protein coat called a capsid. In order to make more viruses, DNA (or RNA) must be synthesized, and so too must proteins. But viruses don't have the 'machinery' needed to do that. Viruses inject their genetic material (DNA or RNA) into the host cell, and it 'hijacks' the host cell's synthetic machinery. The host cell's ribosomes - along with the host cell's tRNA's, amino acids, GTP molecules, eloggation factors, and so on) - synthesize new proteins that will make up the viral capsids; and the host cell's RNA primase, polymerases, free nucleotides, etc. synthesize new DNA (or RNA) that will make up the hundreds or thousands of new virus's genomes.