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3.7 – Evaluate Determinants and Apply Cramer’s Rule

Learn how to evaluate determinants of square matrices and apply Cramer's rule to solve linear systems. Examples and applications included.

emmanuel-manning
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3.7 – Evaluate Determinants and Apply Cramer’s Rule

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  1. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Associated with each square (n x n) matrix is a real number called its determinant. The determinantof a matrix A is denoted by det A or by |A|. The determinant of a 2 x 2 matrix is the difference of the products of the elements of the diagonals.

  2. 3.7 – Evaluate Determinants and Apply Cramer’s Rule

  3. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Example 1: Evaluate the determinant of the matrix.

  4. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Example 2: Evaluate the determinant of the matrix.

  5. 3.7 – Evaluate Determinants and Apply Cramer’s Rule You can use a determinant to find the area of a triangle whose vertices are points in a coordinate plane.

  6. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Example 3: Off the coast of California lies a triangular region of the Pacific Ocean where huge populations of sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Ano Nuevo Island, as shown (next slide). Use the determinant to estimate the area of the region.

  7. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Example 3: Units measured in miles.

  8. 3.7 – Evaluate Determinants and Apply Cramer’s Rule You can use determinants to solve a system of linear equations. The method, called called Cramer’s rule and named after the Swiss mathematician Gabriel Cramer, uses the coefficient matrix of the linear system.

  9. 3.7 – Evaluate Determinants and Apply Cramer’s Rule

  10. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Example 4: Use Cramer’s rule to solve the system:

  11. 3.7 – Evaluate Determinants and Apply Cramer’s Rule

  12. 3.7 – Evaluate Determinants and Apply Cramer’s Rule Example 5: Use Cramer’s rule to solve the linear system.

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