Cramer's Rule and Determinants

Front

What is Cramer's Rule?

Back

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 a determinant?

Back

A determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix.

Front

How do you calculate the determinant of a 2x2 matrix?

Back

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

Front

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

Back

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

Front

What is the condition for a system of equations to have a unique solution using Cramer's Rule?

Back

The determinant of the coefficient matrix must be non-zero.

Front

What does it mean if the determinant of a matrix is zero?

Back

If the determinant is zero, the matrix is singular, indicating that the system of equations may have no solutions or infinitely many solutions.

Front

Solve using Cramer's Rule: 2x + 3y = 6, 4x + 6y = 12

Back

No unique solution (the equations are dependent).

Front

Solve using Cramer's Rule: x + 2y = 5, 2x + 4y = 10

Back

No unique solution (the equations are dependent).

Front

What is the first step in applying Cramer's Rule?

Back

Calculate the determinant of the coefficient matrix.

Front

How do you find the value of a variable using Cramer's Rule?

Back

Replace the corresponding column of the coefficient matrix with the constant terms and calculate the determinant, then divide by the determinant of the coefficient matrix.