2 x 2 Matrices, Determinants, and Inverses

 Definition 1: A square matrix is a matrix with the same number of columns and rows.  Definition 2: For an n x n square matrix, the multiplicative identity matrix is an n x n square matrix I, or I n, with 1’s along the main diagonal and 0’s elsewhere.

 If A and X are n x n matrices, and AX = XA = I, then X is the multiplicative inverse of A, written A -1.

 Show that the matrices are multiplicative inverses.

 Definition 4: The determinant of a 2 x 2 matrix is ad – bc.

 Evaluate each determinant.

 Let. If det A = 0, then A has no inverse.  If det A ≠ 0, then

 Determine whether each matrix has an inverse. If an inverse matrix exists, find it.

AX = B A - 1(AX) = A -1 B (A -1 A)X = A -1 B IX = A -1 B X = A -1 B

 Solve each matrix equation in the form AX = B.

 Communications The diagram shows the trends in cell phone ownership over four consecutive years.  Write a matrix to represent the changes in cell phone use.  In a stable population of 16,000 people, 9927 own cell phones, while 6073 do not. Assume the trends continue. Predict the number of people who will own cell phones next year.  Use the inverse of the matrix from part (a) to find the number of people who owned cell phones last year.