Find the Variables [[a,b],[c,d]]=det[[a,b],[c,d]]

Problem

[[a,b],[c,d]]=det(a)

Solution

1. Identify the components of the equation. The left side is a 2×2 matrix, and the right side is the determinant of that same matrix.

2. Calculate the determinant of the 2×2 matrix using the formula det(a)=a*d−b*c

3. Set up the equality between the matrix and the scalar value. For a matrix to equal a scalar, every entry in the matrix must equal that scalar.

[[a,b],[c,d]]=a*d−b*c

4. Formulate a system of equations based on the equality of the corresponding entries.

a=a*d−b*c

b=a*d−b*c

c=a*d−b*c

d=a*d−b*c

5. Observe that all entries must be equal to each other because they all equal the same determinant value.

a=b=c=d

6. Substitute the equality a=b=c=d back into the determinant equation.

a=a(a)−a(a)

a=a2−a2

a=0

7. Conclude that since a=0 and all variables are equal, all variables must be zero.

a=b=c=d=0

Final Answer

[[a,b],[c,d]]=det(a)⇒a=0,b=0,c=0,d=0

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