13. Multiplication of determinants

Consider two matrices, each with n rows n and columns:

The inner product of the i-th row of A and the k-th column of B is defined by

c_ik = a_i1b_k1 + a_i2b_k2 + ··· + a_inb_kn.^* (13.1)

* If you interpret the rows of A and B as vectors, you have the scalar or inner product used in the vector calculus.

We form the matrix

and compute the determinant C. For this purpose, we treat the a_ik as variables, the b_ik as constants and consider |C| in its dependence on the a_ik. From I in Section 7, applied to |C|, and (13.1) follows that |C| is linear and homogeneous with respect to the elements of each row of A. If you interchange in the matrix A the i-th row with the j-th row, the c_ik take the place of the c_jk (k =1,2,···,n), that is, there also occurs in the matrix C an interchange of the i-th and j-th rows. By II of Section 7, applied to |C|, you find: |C| changes its sign when two rows in the matrix A are interchanged. In particular, if you set a₁₁= ··· = a_nn = 1 and a_ik = 0 for i k, that is, A = E, then c_jk = b_ki and |C| = |B|, whence follows from the comment regarding Rule 3 in Section 8 that

|C| = |A|·|B|.

Corresponding results arise when A or B are replaced by the transposed matrices and one takes into account that in the process |A| and |B| do not change. Hence there arise four possibilties of forming the product of determinants. If you write instead of (13.1)

you obtain again C| = |A|·|B|.

Multiplication rule for determinants:

If the elements of the matrix C are the inner products of the rows or columns of A with the rows or columns of B, then

|C| = |A|·|B|.

Exercises

9. What is the value of a determinant in which there are above (below) the diagonal from the top left to the bottom right zeroes?

10. Compute the value of the n-th order determinant

11. Compute

12. Compute .

13. A skew-symmetric determinant (a_ik = -a_ki) of odd order is zero.

Answers

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