Warm-up A= B= C= D=   2 0 3 -5 -1 4 -7 2 1 6 0 -6  2 9 4  -3   1. Find 8A 2. Find AC 3. Find CD 2 -1  4. Find BD 2x2 Matrices, Determinants, and Inverses Goal To evaluate determinants and inverses of 2x2 matrices and to use inverse matrices to solve equations Thinking Skill To make decisions after reflection and review Definitions  Square Matrix  Matrix with the same number of rows as columns Definitions  Multiplicative Identity Matrix  For an nxn matrix, the multiplicative identity matrix is an nxn square matrix I, or Inxn, with 1s along the main diagonal and 0s elsewhere. Definitions  Multiplicative Inverse of a Matrix  If A and X are nxn matrices, and AX = XA = I, then X is the multiplicative inverse of A written as A-1      Show that the matrices are multiplicative inverses -2 -5 -8 5 -3 -8 & 3 -2 Definitions  Determinant of a 2x2 matrix  a c  The determinant of a 2x2 matrix  is ad - bc  b d   Notation det A  EX  a c b d Evaluate the determinant 7 8  det    5 9   a  b  det   b a   Definitions  Inverse of a 2x2 matrix  a b Let A =   . If det A  0, then A has an inverse.  If det A 0, then c d  1 d  b  1 d  b  A       det A  c a  ad  bc  c a  1 Determine if it has an inverse…if so, find it [flip a and d, make b and c negative] 1 d  b  1 d  b  A     det A  c a  ad  bc  c a  1 6 5  25 20   12 4 9 3    Using Inverse Matrices to solve equations  AX = B A-1(AX) = A-1B (A-1A)X = A -1B IX = A-1B X = A-1B Solving a Matrix Equation 3  4 0  22 4  5  X  0  28      Steps     Get in form AX = B Find A-1 Use X = A-1B Check Lesson Quiz 1. 2. 3. 4. 5 2  2 1  the   Is know? 1 2 inverse of  ? How do you  2 5  Find the determinant of Find the inverse Solve  12 5   16 4    2 4  of 3   7  20 35  3 1 2  X  7     