In a data–processing pipeline

MATHEMATICS · Subjective · CBSE · Mathematics · Matrices · Matrices

Problem

In a data–processing pipeline, a column vector of

5

measurements is first transformed by a matrix

T_{1}

of order

6\times 5

, then by a matrix

T_{2}

of order

4\times 6

, and finally by a matrix

T_{3}

of order

5\times 4

. (a) For each of the following compositions, state whether it is possible. Whenever it is possible, write the order of the resulting matrix (do not perform any numerical multiplication): (i)

T_{2}T_{1}

(ii)

T_{3}T_{2}

(iii)

T_{3}T_{2}T_{1}

(iv)

T_{1}T_{2}

. (b) The engineer wishes to replace the three–step process by a single matrix that acts directly on the original

5\times 1

data vector and produces the same final output as

T_{3}T_{2}T_{1}

. Explain whether this is possible and, if so, state the order of that single matrix. (c) Justify why the composition

T_{1}T_{3}

is not meaningful in this context, even though

T_{1}

and

T_{3}

are individually defined matrices.

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