Complete the proof of Cramer's Rule for two equations...

Complete the proof of Cramer's Rule for two equations containing two variables. [Hint: In system (5), if $a-0$, then $b \neq 0$ and $c \neq 0$, since $D=-b c \neq 0$. Now show that equation (6) provides a solution of the system when $a-0$. Then three cases remain: $b=0, c-0$, and $d=0$.

Complete the proof of Cramer's Rule for two equations containing two variables.
[Hint: In system (5), if $a-0$, then $b \neq 0$ and $c \neq 0$, since $D=-b c \neq 0$. Now show that equation (6) provides a solution of the system when $a-0$. Then three cases remain: $b=0, c-0$, and $d=0$.

Key Concepts

Cramer's Rule

Cramer's Rule is a theorem in linear algebra that provides an explicit formula for solving a system of linear equations with as many equations as unknowns using determinants. The rule states that if the determinant of the coefficient matrix is non-zero, then each variable can be solved for by replacing the corresponding column of the coefficient matrix with the constants from the right-hand side of the equations, computing the determinant of this new matrix, and dividing by the original determinant.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important insights about the matrix, including whether it is invertible and the volume transformation associated with the linear map it represents. In the context of a 2x2 matrix, the determinant is calculated as ad - bc, and a non-zero determinant indicates that the matrix has full rank, which is a key requirement for applying Cramer's Rule.

Linear Systems of Equations

A system of linear equations is a collection of one or more linear equations involving the same variables. The solution of the system is the set of values for the variables that satisfies all the equations simultaneously. Techniques for solving these systems include substitution, elimination, matrix operations, and, as in Cramer's Rule, the use of determinants. The uniqueness of the solution is directly related to the non-zero determinant of the coefficient matrix.

Proof by Cases

Proof by Cases is a method of mathematical proof that involves breaking a problem into several separate cases, each of which is analyzed individually. This approach is especially useful when the problem's structure or the conditions imposed on the variables produce distinct scenarios, such as when certain coefficients are zero. By confirming that the result holds under each circumstance, one can establish the general validity of the theorem or rule being proved.

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