Determinants & Cramer's Rule

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What is a determinant in mathematics?

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A determinant is a scalar value that is a function of the entries of a square matrix. It provides important information about the matrix, such as whether it is invertible.

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

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For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is calculated as \( det(A) = ad - bc \).

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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.

Tags

CCSS.HSA.REI.C.9

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When can Cramer's Rule be applied?

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Cramer's Rule can be applied when the system of equations has a unique solution, which occurs if the determinant of the coefficient matrix is non-zero.

Tags

CCSS.HSA.REI.C.9

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What is the determinant of a 3x3 matrix?

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For a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \), the determinant is calculated as \( det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).

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

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If the determinant of a matrix is zero, it indicates that the matrix is singular, meaning it does not have an inverse and the system of equations may have either no solution or infinitely many solutions.

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How do you find the inverse of a 2x2 matrix using determinants?

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For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the inverse is given by \( A^{-1} = \frac{1}{det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \) if \( det(A) \neq 0 \).

Tags

CCSS.HSA.REI.C.9

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What is the relationship between determinants and linear transformations?

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The determinant of a matrix representing a linear transformation indicates the scaling factor of the transformation and whether it preserves orientation.

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How can determinants be used to find the area of a triangle?

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The area of a triangle formed by points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) can be found using the determinant: \( Area = \frac{1}{2} \left| det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right| \).

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What is the cofactor of an element in a matrix?

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The cofactor of an element \( a_{ij} \) in a matrix is given by \( C_{ij} = (-1)^{i+j} det(M_{ij}) \), where \( M_{ij} \) is the submatrix formed by deleting the i-th row and j-th column.