5 minutes Warm-Up Multiply Matrix A times Matrix B.

Objectives: Find and use the determinant of a matrix 4.6 The Inverse of a Matrix Objectives: Find and use the determinant of a matrix

Identity Matrix The Identity matrix is the matrix where the diagonal starting with the first entry going down to the last entry is filled with 1’s and all other entries are 0’s. When you multiply a matrix by it’s inverse you get the identity matrix. To check and see if two matrices are inverses multiply them together and see if you get the identity matrix.

The Inverse of a Matrix AB = I You can use the equation AB = I to find the inverse of a matrix. AB = I 1(a) + 2(c) 1(b) + 2(d) 3(a) + 5(c) 3(b) + 5(d) a + 2c = 1 b + 2d = 0 3a + 5c = 0 3b + 5d = 1

The Inverse of a Matrix You can use the equation AB = I to find the inverse of a matrix. (-3) ( ) ( ) a + 2c = 1 (-3) (-3) ( ) ( ) b + 2d = 0 (-3) 3a + 5c = 0 3b + 5d = 1 -3a - 6c = -3 -3b – 6d = 0 3a + 5c = 0 3a + 5d = 1 c = 3 d = -1 a = -5 b = 2

The Determinant of a Matrix The determinant of A, denoted by det(A) is defined as Matrix A has an inverse iff

Example 1 AB = I Find the determinant, tell whether the matrix has an inverse, and find the inverse (if it exists). det(A) = ad - bc = (2)(2) – (3)(1) = 1 , so matrix A has an inverse AB = I To find the inverse of the matrix simply type in your matrix and then select the matrix you want to take the inverse of and hit the inverse button on the calculator.

Example 1 Find the determinant, tell whether the matrix has an inverse, and find the inverse (if it exists). 2(a) + 3(c) 1(a) + 2(c) 2(b) + 3(d) 1(b) + 2(d) 2a + 3c = 1 2b + 3d = 0 a + 2c = 0 b + 2d = 1 a = -2c b = 1 – 2d 2(-2c) + 3c = 1 2(1 – 2d) + 3d = 0 -4c + 3c = 1 2 – 4d + 3d = 0 c = -1 2 – d = 0 a = 2 d = 2 b = -3

Practice Find the determinant, and tell whether each matrix has an inverse. 1) 2)

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