18.5 Matrix Algebra 613
Figure 18.15. Then, we add the products of the diagonal elements lying on
the solid arrows and subtract them from the products of the diagonal ele-
ments lying on the dashed arrows. This procedure shown in Figure 18.15
results in the determinant value given by Equation (18.17).
The direct-expansion procedure cannot be used to obtain higher-
order determinants. Instead, we resort to a method that first reduces the order
of the determinant — to what is called a minor — and then evaluates the
lower-order determinants. You will learn about minors later in your other
classes.Example 18.3 Given the following matrix: , calculate the determinant of [A].
As explained earlier, using the direct-expansion method, we repeat and place the first
and the second column of the matrix next to the third column as shown, and compute
the products of the elements along the solid arrows, then subtract them from the prod-
ucts of elements along the dashed arrows, as shown in Figure 18.16. Use of this method
results in the following solution.When the determinant of a matrix is zero, the matrix is called asingular. A singular matrix
results when the elements in two or more rows of a given matrix are identical. For example,consider the following matrix: , whose rows one and two are identical.
As shown next, the determinant of [A] is zero.
Matrix singularity can also occur when the elements in two or more rows of a matrix are
linearly dependent. For example, if we multiply the elements of the second row of matrix [A]bya scalar factor such as 7, then the resulting matrix is singular because
rows one and two are now linearly dependent. As shown next, the determinant of the new
[A] matrix is zero.†
21 4
14728
13 5
† 122172152 1121282112 1421142132
3 A 4 £
21 4
14728
13 5
§
†
214
214
135
† 122112152 112142112 142122132
3 A 4 £
214
214
135
§
†
150
837
6 29
† 112132192 152172162 1021821 22
3 A 4 £
150
837
6 29
§
1 5 0 1 5
8 3 7 8 3
6 2 9 6 2
■Figure 18.16
The direct expansion method
for Example 18.3.a 11 a 12
a 21 a 22
(a)(b)c 11 c 12 c 13 c 11 c 12
c 21 c 22 c 23 c 21 c 22
c 31 c 32 c 33 c 31 c 32
■Figure 18.15
Direct-expansion procedure for computing the determinant of
(a) 2 2 matrix, and (b) 3 3 matrix.152182192 1121721 22 102132162 109
112122152 122142132 142112112 0
1121142152 1221282132 142172112 0
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