Simplify the Matrix

Problem

[[1,−1],[−2,1],[3,1]]*[[15/14],[4]]−[[1],[5],[8]]

Solution

1. Multiply the 3×2 matrix by the 2×1 column vector using the dot product of rows and columns.

[[1,−1],[−2,1],[3,1]]*[[15/14],[4]]=[[(1)*(15/14)+(−1)*(4)],[(−2)*(15/14)+(1)*(4)],[(3)*(15/14)+(1)*(4)]]

2. Calculate the individual entries of the resulting product vector.

[[15/14−56/14],[−15/7+28/7],[45/14+56/14]]=[[−41/14],[13/7],[101/14]]

3. Subtract the second vector from the result of the multiplication by finding common denominators.

[[−41/14],[13/7],[101/14]]−[[1],[5],[8]]=[[−41/14−14/14],[13/7−35/7],[101/14−112/14]]

4. Simplify the final vector components.

[[−55/14],[−22/7],[−11/14]]

Final Answer

[[1,−1],[−2,1],[3,1]]*[[15/14],[4]]−[[1],[5],[8]]=[[−55/14],[−22/7],[−11/14]]

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