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1.10 Evaluate Determinants and Apply Cramer’s Rule

Explore the concept of inverses in mathematics, including determinants, Cramer’s Rule, and identity matrices. Learn how to find inverses of matrices and their importance.

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1.10 Evaluate Determinants and Apply Cramer’s Rule

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  1. 1.10 Evaluate Determinants and Apply Cramer’s Rule Pg. 234

  2. Inverses What does “inverse” mean to you? What is the inverse of multiplication?

  3. Identity What does “identity” mean to you? What is the multiplicative identity for the real numbers? In other words, 5 * __= 5? The identity for multiplication is 1 because anything multiplied by 1 will be itself.

  4. What do we multiply by to get the identity? In other words, 5 * ___=1? a * a-1= 1 Any number multiplied by its inverse will be the identity.

  5. Identity Matrix Any matrix multiplied by its inverse will be the identity matrix. A * A-1= I A-1 *A = I 3x3 Identity Matrix 2x2 Identity Matrix

  6. Identity Matrix Just like 5*1 = 5… AI= A IA= A Or

  7. Determine whether A and B are inverses. Find out if AB = I If so, then yes they’re inverses. If not, then no they’re not inverses. YES

  8. Determine whether A and B are inverses. YES

  9. Determine whether A and B are inverses. NO

  10. In order to find the inverse….. First, we need a determinant. = ad – bc As long as ad – cb ≠ 0 If ad – cb = 0, then the matrix has no inverse!!!! (VERY IMPORTANT)

  11. Determinant of a 3 x 3 Matrix

  12. Find the Inverse of a 2 x 2 Formula A-1= determinant

  13. Find A-1, if it exists. First find the Determinant If it exists, put into the formula and find the inverse A-1=

  14. Find A-1, if it exists. A-1=

  15. Ex. 6 Find A-1, if it exists. Does not exist, because it’s not square. You can only find the Inverse of a Square Matrix

  16. Ex. 7 Find A-1, if it exists.

  17. Questions????

  18. Assignment Pg. 51 1 – 17 odd Pg. 52 1 – 19 odd (3x3 or larger use calculator)

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