Simplify the Matrix -a+b[[-2,2,8],[2,-1,-2],[2,-1,-4]]

Problem

−a+b*[[−2,2,8],[2,−1,−2],[2,−1,−4]]

Solution

1. Distribute the scalar b to every element inside the matrix.

b*[[−2,2,8],[2,−1,−2],[2,−1,−4]]=[[−2*b,2*b,8*b],[2*b,−b,−2*b],[2*b,−b,−4*b]]

2. Rewrite the expression by combining the scalar −a with the resulting matrix. Since −a is a scalar being added to a matrix, it is typically interpreted in linear algebra as −a*I where I is the identity matrix of the same dimension.

−a*[[1,0,0],[0,1,0],[0,0,1]]+[[−2*b,2*b,8*b],[2*b,−b,−2*b],[2*b,−b,−4*b]]

3. Perform matrix addition by adding the corresponding elements of the two matrices.

[[−a,0,0],[0,−a,0],[0,0,−a]]+[[−2*b,2*b,8*b],[2*b,−b,−2*b],[2*b,−b,−4*b]]

4. Simplify each entry to reach the final matrix form.

[[−a−2*b,2*b,8*b],[2*b,−a−b,−2*b],[2*b,−b,−a−4*b]]

Final Answer

−a+b*[[−2,2,8],[2,−1,−2],[2,−1,−4]]=[[−a−2*b,2*b,8*b],[2*b,−a−b,−2*b],[2*b,−b,−a−4*b]]

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