Assignment 5   -   Solutions

Find the determinant and the inverse of the matrix

Find the determinant of the matrix

A row operation could be used to reduce this determinant to triangular form, but a cofactor expansion along the last row or column is much easier:

[All matrices representing pure rotations have a determinant of 1.]

Find the determinant of the matrix

Use row operations to transform the determinant into upper triangular form:


OR one may use a cofactor expansion, starting with any of column 1 or row 2 or row 4.
Expanding along row 4:

Find the determinant of the matrix

Columns 1 and 4 of this determinant are identical.
Immediately we have

[Textbook, page 114, exercises 3.1, question 8(b)]
Show that  

Try to eliminate the " 2a " term from element (1,1) by the row operation R₁ – 2 R₃.
However, this introduces an unwanted new term " –4x ", which can be eliminated in turn by the row operation R₁ + 4 R₂.
The two row operations together leave a multiple of " p " alone in the (1,1) element.





Therefore

Find the value(s) of   c   for which the matrix   F   is singular.

Use a cofactor expansion along row 3 (column 3 also works well):


OR one may reduce the determinant to triangular form:


F   is singular if and only if   det F = 0 ; that is

If   det X = 2   and   det Y = 3 , then calculate the value of
  where possible.
Under what circumstances does     not exist?.

Provided the two matrices   X   and   Y   are of the same dimensions as each other, all of the products in     exist.
The fact that   det Y   is not zero guarantees that   Y ^-1   exists.
Using the results

The product     does not exist if and only if the square matrices   X   and   Y   have different dimensions.

The general 3×3 skew-symmetric matrix is
  (where   a, b, c   are any real numbers).
Find   det K.

The determinant may be calculated by a cofactor expansion along any row or down any column.
Arbitrarily choosing row 1:

OR one may reduce the determinant to triangular form:


There is a row of all zero entries.   Therefore, for all values of   a, b, c ,

and all 3×3 skew-symmetric matrices are singular.
Note that the same is not true for 2×2 skew-symmetric matrices.