Simplify the Matrix [[1,0,1],[1,2,2],[0,3,3]][[0],[0],[0]]

Problem

[[1,0,1],[1,2,2],[0,3,3]]*[[0],[0],[0]]

Solution

1. Identify the dimensions of the matrices. The first matrix is a 3×3 matrix and the second is a 3×1 column vector.

2. Apply the rule for matrix-vector multiplication, where each element of the resulting 3×1 vector is the dot product of a row from the first matrix and the column vector.

3. Calculate the first entry:

(1⋅0)+(0⋅0)+(1⋅0)=0

4. Calculate the second entry:

(1⋅0)+(2⋅0)+(2⋅0)=0

5. Calculate the third entry:

(0⋅0)+(3⋅0)+(3⋅0)=0

6. Conclude that multiplying any matrix by a zero vector of compatible dimensions results in a zero vector.

Final Answer

[[1,0,1],[1,2,2],[0,3,3]]*[[0],[0],[0]]=[[0],[0],[0]]

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