A 3x3 matrix A is defined as
| A= | a11 | a12 | a13 |
| a21 | a22 | a23 | |
| a31 | a32 | a33 |
1. Write a program which
a) calculates and prints on the screen the determinant of matrix A, i.e. a scalar det(A) defined as:
det(A) = a11 * a22 * a33 + a13 * a21 * a32 + a12 * a23 * a31 - a13 * a22 * a31 - a12 * a21 * a33 - a11 * a23 * a32
b) calculates the inverse of matrix A (if it exists), which is
A-1 = (1 / det(A)) * BT, if det(A) != 0
where
| B= | b11 | b12 | b13 |
| b21 | b22 | b23 | |
| b31 | b32 | b33 |
with elements
b11 = a22 * a33 - a23 * a32, b12 = a23 * a31 - a21 * a33, b13 = a21 * a32 - a22 * a31,
b21 = a13 * a32 - a12 * a33, b22 = a13 * a33 - a13 * a31, b23 = a12 * a31 - a11 * a32,
b31 = a12 * a23 - a13 * a22, b32 = a13 * a21 - a11 * a23, b33 = a11 * a22 - a12 * a21
The matrix BT
| B= | b11 | b12 | b13 | = | b11 | b21 | b31 |
| b21 | b22 | b23 | b12 | b22 | b32 | ||
| b31 | b32 | b33 | b13 | b23 | b33 |
is the transpose of the matrix B. Note that the product of a scalar "a" and some matrix D is a matrix
| a * D = a * | d11 | d12 | d13 | = | a * d11 | a * d12 | a * d13 |
| d21 | d22 | d23 | a * d21 | a * d22 | a * d23 | ||
| d31 | d32 | d33 | a * d31 | a * d32 | a * d33 |
Note, that if , the inverse of the matrix A does not exist, and the matrix A is called not invertible or singular or degenerate
c) writes into the file the matrix A-1 (if it exists).
d) checks that the matrix product A-1 * A = I, where
| I = | 1 | 0 | 0 |
| 0 | 1 | 0 | |
| 0 | 0 | 1 |
is a 3x3 identity matrix, or unit matrix. Your programme should print on the screen the elements of the matrix A-1 * A. Note that the matrix product of two matrices A and B is a matrix C ( )
| C= | c11 | c12 | c13 |
| c21 | c22 | c23 | |
| c31 | c32 | c33 |
whose elements are
| 3 | |
| cij= | E aij * b kj, i, j, k = 1, 2, 3. |
| k=1 |
2. Test the program for
| A= | 2.0 | 0 | 0 |
| 0 | 4.0 | 0 | |
| 0 | 0 | 6.0 |
| A= | 1.0 | 2.0 | 3.0 |
| 4.0 | 5.0 | 6.0 | |
| 7.0 | 8.0 | 9.0 |