Instruction

1

There are three operations on matrices: addition, subtraction and multiplication. The division of the matrices as such action is not, but it can be represented as a multiplication of the first

**matrix**to the matrix inverse to the second:A/B = A·B^(-1).2

*Therefore, the operation of division of matrices is reduced to two steps: finding the inverse matrix and multiply it by the first. The inverse is the matrix A^(-1), which when multiplied by A yields the identity matrix.*

3

The formula of inverse matrix: A^(-1) = (1/∆)•B, where ∆ is the determinant of the matrix that should be nonzero. If not, then the inverse matrix does not exist. B – matrix from algebraic additions of the original matrix A.

4

*For example, do the division of the given matrices.*

5

Find the inverse of the second. To do this, compute its determinant and the matrix of algebraic additions. Write down the formula of the determinant of a square matrix of the third order:∆ = a11·a22·a33 + a12·a23·a31 + a21·a32·a13 – a31·a22·a13 – a12·a21·a33 – a11·a23·a32 = 27.

6

Define algebraic additions to these formulas:A11 = a22•a33 - a23•a32 = 1•2 – (-2)•2 = 2 + 4 = 6;A12 = -(a21•a33 - a23•a31) = -(2•2 – (-2)•1) = -(4 + 2) = -6;A13 = a21•a32 - a22•a31 = 2•2 – 1•1 = 4 – 1 = 3;A21 = -(a12•a33 - a13•a32) = -((-2)•2 - 1•2) = -(-4 - 2) = 6;A22 = a11•a33 - a13•a31 = 2•2 – 1•1 = 4 – 1 = 3;A23 = -(a11•a32 - a12•a31) = -(2•2 – (-2)•1) = -(4 + 2) = -6;A31 = a12•a23 - a13•a22 = (-2)•(-2) – 1•1 = 4 – 1 = 3;A32 = -(a11•a23 - a13•a21) = -(2•(-2) - 1•2) = -(-4 - 2) = 6;A33 = a11•a22 - a12•a21 = 2•1 – (-2)•2 = 2 + 4 = 6.

7

*Divide the elements of the matrix algebraic additions to the value of the determinant is equal to 27. So you got the matrix inverse to the second. Now the problem is reduced to the multiplication of the first matrix to the new one.*

8

*Perform the matrix multiplication according to the formula C = A*B:c11 = a11•b11 + a12•b21 + a13•b31 = 1/3;c12 = a11•b12 + a12•b22 + a13•b23 = -2/3;c13 = a11•b13 + a12•b23 + a13•b33 = -1;c21 = a21•b11 + a22•b21 + a23•b31 = 4/9;c22 = a21•b12 + a22•b22 + a23•b23 = 2/9;c23 = a21•b13 + a22•b23 + a23•b33 = 5/9;c31 = a31•b11 + a32•b21 + a33•b31 = 7/3;c32 = a31•b12 + a32•b22 + a33•b23 = 1/3;c33 = a31•b13 + a32•b23 + a33•b33 = 0.*