Square Matrix

1. Square Matrix A square matrix is a matrix with the same number of columns as rows.

2. 2 X 2 Matrices, Determinants and Inverses Multiplicative Identity Matrix For an n x n matrix, the multiplicative identity is an n x n matrix I, or In x n, with 1’s along the main diagonal and 0’s elsewhere. I2 = I3 = • 0 0 • 0 1 0 • 0 0 1 • 0 • 0 1 and so forth

3. Multiplicative Inverse of a Matrix If A and X are n x n matrices, and AX =XA=I, then X is the multiplicative inverse of A, written as A-1. AA-1 = A-1A = I

4. 1 • 2.5 1 -2 2 5 -4 and -2 -5 -3 -8 -8 5 3 -2 and 3 -1 7 1 -0.1 0.1 -0.7 0.3 and Verifying Inverses Show that the matrices are multiplicative inverses. 1. 2. 3. Yes Yes No

5. a b c d a b c d Determinant of a 2 x 2 Matrix The determinant of a 2 x 2 matrix is ad – bc. Symbols for the determinant of a matrix: det A

6. 2 • 4 2 • 2 • 4 2 7 8 -5 -9 • -3 • 5 6 k 3 3 – k -3 Evaluating the Determinant of a 2 x 2 Matrix Evaluate each determinant. 1. 2. 3. 4. = 4(2) – 4(2) = 0 = 7(-9) – (-5)(8) = -23 = 4(6) – (5)(-3) = 39 = k(-3) – (3-k)(3) = -3k – (3 – 3k) = -3k – 3 + 3k = -3

7. a b c d d -b -c a Finding an Inverse Inverse of a 2 x 2 Matrix Let A = . If det A  0, then A has an inverse. If det A  0, then A -1 =

8. 2 4 1 3 3 -4 -1 2 3/2 -2 -1/2 1 M = Determine whether each matrix has an inverse. If an inverse exists, find it. 1. Solution: Find the determinant: det M = 2(3) – 1(4) = 2 Since the determinant of M is not equal to 0, then the inverse exists. Find the Inverse M-1 = =

9. -24/11 23/33 10/11 -5/33 2/27 4/9 10/27 2/9 -4 1 5 -1.2 0.5 2.3 3 7.2 12 4 9 3 6 5 25 20 -1.5 3 2.5 -0.5 Yes; 2. 3. 4. 5. No Yes; Yes;