Find the Null Space det [[1,2,3],[1,4,5],[y,4,5]]=0

Problem

det(1)=0

Solution

1. Calculate the determinant of the matrix using cofactor expansion along the first row.

det(1)=1*(4⋅5−5⋅4)−2*(1⋅5−5⋅y)+3*(1⋅4−4⋅y)

2. Simplify the terms inside the parentheses.

det(=)*1*(20−20)−2*(5−5*y)+3*(4−4*y)

3. Distribute the constants into the simplified terms.

det(=)*0−10+10*y+12−12*y

4. Combine like terms to find the final expression for the determinant.

det(=)*2−2*y

5. Set the determinant equal to zero as specified in the task and solve for y

2−2*y=0

−2*y=−2

y=1

Final Answer

det(1)=0⇒y=1

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