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Section 4.4

Section 4.4. Identity and Inverse Matrices. Identity matrix. The Multiplicative identity is 1 because a x 1 = a and 1 x a = a. The identity matrix of an nxn matrix is an nxn matrix that has 1’s on its main diagonal and 0’s everywhere else. 2x2 identity matrix:. Identity matrix.

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Section 4.4

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  1. Section 4.4 Identity and Inverse Matrices

  2. Identity matrix • The Multiplicative identity is 1 because a x 1 = a and 1 x a = a. • The identity matrix of an nxn matrix is an nxn matrix that has 1’s on its main diagonal and 0’s everywhere else. • 2x2 identity matrix:

  3. Identity matrix • 3x3 identity matrix: • So any matrix multiplied by the identity matrix is itself!

  4. Inverse Matrices • Two nxn matrices are inverses if their product (in both orders) is the nxn identity matrix. • For example: • If AB = I and BA = I, then B = A-1

  5. Example • Are and inverses? • Check: AB and BA • AB: = • BA: = • Therefore, they are inverses.

  6. Finding the inverse • The inverse of matrix A = is

  7. Find the inverse of • A = a=3, b=1, c=4, d=2

  8. Find the inverse of • a=4, b=-5, c=-3, d=4

  9. Solving Matrix Equations • Solve AX = B for the following: A X = B • First, find A-1. • Then multiply both sides of the equation by A-1 on the left.

  10. Find the inverse of • a=4, b=-1, c=-3, d=1

  11. Multiply both sides of equation by A-1 on the left. A-1AX = A-1B

  12. Use a calculator to find the inverse of A. Use the calculator to verify the results. Type in matrix values in calculator Then, 2nd matrix, X-1, enter

  13. Verify: • Does • If so, inverse is correct • If not, inverse is not correct

  14. Find the inverse of matrix A and verify:

  15. Assignment Section 4.4: page 227-229 # 14-20 even, 25, 26, 33 – 39, 51 – 52

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