Solving systems using Cramer's Rule

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What is Cramer's Rule?

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Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.

Front

What is the determinant of a 2x2 matrix?

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For a 2x2 matrix [[a, b], [c, d]], the determinant is calculated as ad - bc.

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How do you set up a system of equations for Cramer's Rule?

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To use Cramer's Rule, express the system of equations in the form Ax = B, where A is the coefficient matrix, x is the variable matrix, and B is the constant matrix.

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What is the formula for finding the solution using Cramer's Rule?

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The solution for each variable x_i is given by x_i = det(A_i) / det(A), where A_i is the matrix formed by replacing the i-th column of A with the constant matrix B.

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What is the significance of the determinant being zero in Cramer's Rule?

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If the determinant of the coefficient matrix A is zero, the system of equations either has no solution or infinitely many solutions.

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How do you calculate the determinant of a 3x3 matrix?

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For a 3x3 matrix [[a, b, c], [d, e, f], [g, h, i]], the determinant is calculated as a(ei - fh) - b(di - fg) + c(dh - eg).

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What is the first step in applying Cramer's Rule to a system of equations?

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The first step is to calculate the determinant of the coefficient matrix.

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What does it mean if a system of equations has a unique solution?

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A unique solution means that there is exactly one set of values for the variables that satisfies all equations in the system.

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How do you interpret the solution (x,y) in the context of a system of equations?

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The solution (x,y) represents the point of intersection of the lines represented by the equations in the system.

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What is the role of the coefficient matrix in Cramer's Rule?

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The coefficient matrix contains the coefficients of the variables in the system of equations and is used to calculate the determinant.