Inverse & Identity Matrices Section 4.5

Objectives You will write the identity matrix for any square matrix find the inverse of a 2 x 2 matrix

The Identity Matrix I The identity matrix is a square matrix with 1’s on the principal diagonal. All other elements are 0 It’s the only matrix that’s commutative A • I = I • A = A Principal Diagonal = If you multiply the identity matrix is by a 2nd matrix, the product is equal to the second matrix (it’s like multiplying by 1) The identity matrix will always be the same dimension as the other matrix.

Inverse Matrix A-1 n • 1/n = 1/n • n = 1 is the multiplicative inverse (for real numbers ≠ 0) A• A-1 = A-1 • A = 1 for matrices if A-1 exists A 2x2 matrix will have an inverse if its determinant ≠ 0 has the determinant = ad – cb If ad = cb, the matrix does not have an inverse

Finding A-1 Switch A1,1 & A 2, 2 Find the determinant of the matrix If A = Switch A1,1 & A 2, 2 Find the determinant of the matrix  = ad – cb If ad – cb = 0, you’re done—no inverse exists Then change the signs on A1,2 & A2,1 & put the fraction on the left If ad ≠ cb the matrix has an inverse Put “1” over the value of the determinant = A-1

find M-1 If M = det of M = Change the matrix to inverse form: = 14 – 0 = 14 The fraction to use for the inverse is Change the matrix to inverse form: (7 & 2 changed places, the sign on the –5 changed to positive & 0 stayed 0) Set up the inverse: = M-1

Gotta get in some practice! Find A-1 if A = (click to check) (click to check) Find B-1 if B = Find C-1 if C = (click to check)

Find A-1 if A =  The determinant: = 4(-1) – 2(3) = -10  = 4(-1) – 2(3) = -10  “Everybody Switch! ” ____1____ determinant = Get the fraction Put it all together! _1_ -10 = = A-1

Find B-1 if B = Since B is a 2x3 matrix and is not a square matrix, It doesn’t have an inverse.

Find C-1 if C = 1. Determinant  = 2 (3) – 1 (6) = 0 C does not have an inverse since the determinant is 0. (Section 4.6 explains why.)