9th class science group Math Ch no .1 Exercise no 1.6 Question no.1 part (v,vi)........
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The Crammer rule is a fundamental principle in linear algebra and matrix theory. It is named after the Swiss mathematician Gabriel Crammer.

Statement of the Rule

The Crammer rule states: "If a matrix A has an inverse, then the system of linear equations Ax = b has a unique solution, given by x = A^1 b."

Explanation

The rule provides a method to solve systems of linear equations using matrix inversion. It states that if the coefficient matrix A has an inverse (denoted by A^1), then the solution to the system can be found by multiplying the inverse matrix by the constant vector b.

Mathematical Derivation

Let's consider a system of linear equations:

Ax = b

Where A is an n x n matrix, x is an n x 1 vector, and b is an n x 1 vector.

If A has an inverse, then we can multiply both sides of the equation by A^1:

A^1 Ax = A^1 b

Using the property of inverse matrices, we get:

x = A^1 b

Implications

The Crammer rule has significant implications in various mathematical fields, including:

1. _Linear Algebra_: Solving systems of linear equations.
2. _Calculus_: Finding derivatives and integrals.
3. _Statistics_: Linear regression and data analysis.
4. _Physics_: Solving problems in mechanics, electromagnetism, and quantum mechanics.

Conclusion

In conclusion, the Crammer rule is a powerful tool in mathematics, providing a method to solve systems of linear equations using matrix