Multiply the Matrices

Problem

[[2,2,k]]*([[1,1,0],[1,0,2],[0,2,−1]]*[[2],[2],[k]])

Solution

1. Multiply the 3×3 matrix by the 3×1 column vector.

[[1,1,0],[1,0,2],[0,2,−1]]*[[2],[2],[k]]=[[(1)*(2)+(1)*(2)+(0)*(k)],[(1)*(2)+(0)*(2)+(2)*(k)],[(0)*(2)+(2)*(2)+(−1)*(k)]]

2. Simplify the resulting column vector.

[[2+2+0],[2+0+2*k],[0+4−k]]=[[4],[2+2*k],[4−k]]

3. Multiply the 1×3 row vector by the simplified 3×1 column vector.

[[2,2,k]]*[[4],[2+2*k],[4−k]]=(2)*(4)+(2)*(2+2*k)+(k)*(4−k)

4. Expand the terms in the expression.

8+4+4*k+4*k−k2

5. Combine like terms to find the final scalar value.

−k2+8*k+12

Final Answer

[[2,2,k]]*([[1,1,0],[1,0,2],[0,2,−1]]*[[2],[2],[k]])=−k2+8*k+12

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