Matrix inverses
- 1. Chapter 2Chapter 2 Systems of Linear EquationsSystems of Linear Equations and Matricesand Matrices Section 2.5Section 2.5 Matrix InversesMatrix Inverses
- 2. What is a Matrix Inverse?What is a Matrix Inverse? The inverse of a matrix isThe inverse of a matrix is comparablecomparable to the reciprocal of a real number.to the reciprocal of a real number. The product of a matrix and itsThe product of a matrix and its identity matrix is always the matrixidentity matrix is always the matrix itself.itself. In other words, multiplying a matrixIn other words, multiplying a matrix by its identity matrix is like multiplyingby its identity matrix is like multiplying a number by 1.a number by 1.
- 3. Multiplicative IdentityMultiplicative Identity The real number 1 is the multiplicativeThe real number 1 is the multiplicative identity for real numbers:identity for real numbers: for any real numberfor any real number aa, we have, we have aa • 1 = 1 •• 1 = 1 • aa == aa In this section, we define aIn this section, we define a multiplicativemultiplicative identity matrixidentity matrix II that has propertiesthat has properties similar to those of the number 1.similar to those of the number 1. We use the definition of this matrixWe use the definition of this matrix II toto find the multiplicative inverse of anyfind the multiplicative inverse of any square matrix that has an inverse.square matrix that has an inverse.
- 4. Identity MatrixIdentity Matrix IfIf II is to be the identity matrix, bothis to be the identity matrix, both of the productsof the products AIAI andand IAIA mustmust equalequal A.A. The identity matrix only exists forThe identity matrix only exists for square matrices.square matrices.
- 5. Examples of Identity MatricesExamples of Identity Matrices
- 6. Determining if Matrices areDetermining if Matrices are Inverses of Each OtherInverses of Each Other Recall that a number multiplied by itsRecall that a number multiplied by its multiplicative inverse yields a product of 1.multiplicative inverse yields a product of 1. Similarly, the product of matrixSimilarly, the product of matrix AA and itsand its multiplicative inverse matrixmultiplicative inverse matrix AA (read “A-(read “A- inverse”) isinverse”) is II, the identity matrix., the identity matrix. So, to prove that two matrices are inversesSo, to prove that two matrices are inverses of each other, show that their product,of each other, show that their product, regardless of the order they’re multiplied, isregardless of the order they’re multiplied, is always the identity matrix.always the identity matrix. 1−
- 7. Example 1Example 1 Prove or disprove that the matrices below areProve or disprove that the matrices below are inverses of each other.inverses of each other. a.)a.) b.)b.) c.)c.) 5 7 3 7 2 3 2 5 and − ÷ ÷ − 1 2 5 2 3 5 3 1 and − − − ÷ ÷ − − − 0 1 0 1 0 1 0 0 2 1 0 0 1 1 0 0 1 0 and ÷ ÷ − ÷ ÷ ÷ ÷− −
- 8. Finding the Inverse of a MatrixFinding the Inverse of a Matrix
- 9. Row Operations on MatricesRow Operations on Matrices
- 10. Example 2Example 2 Find the inverse, if it exists, for eachFind the inverse, if it exists, for each matrix.matrix. a.)a.) b.)b.) c.)c.) 1 2 3 4 − − ÷ 5 10 3 6 ÷ − − 1 2 2 1 − ÷ − −
- 11. Shortcut for Finding the Inverse ofShortcut for Finding the Inverse of a 2 x 2 Matrixa 2 x 2 Matrix If a matrix is of the formIf a matrix is of the form then the inverse can be found bythen the inverse can be found by calculating:calculating: Note: ad – bc ≠ 0.Note: ad – bc ≠ 0. a b c d ÷ 1 d b c aad bc − ÷ −−
- 12. Example 3Example 3 Find the inverse of the matrix below usingFind the inverse of the matrix below using the shortcut method.the shortcut method. 4 2 5 3 ÷
- 13. Solution to Example 3Solution to Example 3 1 d b c aad bc − ÷ −− 4 2 5 3 ÷ To find the inverse of the matrix use the formula and simplify. 3 21 5 44(3) 2(5) − ÷ −−
- 14. Solution to Example 3 (continued)Solution to Example 3 (continued) 1 3 21 5 44(3) 2(5) 3 21 5 42 3 1 2 5 2 2 A− − = ÷ −− − = ÷ − − ÷ = ÷ − ÷ ÷