Math Help for Section 8.5, Page 417

The
Determinant of a Matrix
One way to evaluate the determinant of a $3\times3$ matrix, called expanding by
minors, allows you to write the determinant of
a $3\times3$ matrix in terms of three $2\times2$
determinants. The minor
of an entry in a $3\times3$ matrix is the determinant
of the $2\times2$ matrix that remains after deletion of the
row and column in which
the entry occurs. Here is an example of the minors in the first row of
a $3\times3$ matrix.

    The signs of the terms
used in expanding by minors follow the alternating
pattern shown in the Study Tip on page 205. For instance, the signs
used to expand by minors
along the second row are $-,\;+,\;-,$ as shown below.

   
$\eqalign{\text{det}(A)=& \left|\begin{matrix}
a_1 & b_1 & c_1\cr \color{red}a_2 &
\color{red}b_2 & \color{red}c_2\cr
a_3 & b_3 & c_3
\end{matrix}\right|  \cr 
=&-{\color{red}a_2}(\text{minor of
}a_2)+{\color{red}b_2}(\text{minor of
}b_2)-{\color{red}c_2}(\text{minor
of }c_2)  
      }$

Similarly, you can expand minors along the first row as
follows.

   
$\text{det}(A)={a_1}(\text{minor of
}a_1)-{b_1}(\text{minor of }b_1)+{c_1}(\text{minor
of }c_1)$

So, the determinant of the matrix in the example above is

   
$\eqalign{\text{determinant}=& {1}(-34)-({-1})(10)+{3}(4)
 \cr  =& -34+10+12 \cr =&
-12.}  $

Example
2: Tip
A zero entry in a matrix will always yield a zero term when expanding
by minors. So, when you are evaluating the determinant of a matrix,
choose to expand along the row or column that has the most zero
entries. According to this rule, you should expand along the second row or first column of matrix A and you should expand along the second column of matrix B.

Technology: Tip
A graphing calculator with matrix
capabilities can be used to evaluate
the determinant of a square matrix.
Consult the user’s guide of your
graphing calculator to learn how
to evaluate a determinant. Use the
graphing calculator to calculate the determinants of the $2\times2$
matrices in Example 1 on page 204. Then evaluate the determinant of the
$3\times3$
matrix in the example above using a
graphing calculator and confirm that it is $-12$.