8.7. Determinant of Matrices http://www.ck12.org
- Find the determinant of the following matrix.
D=
4 8 3
0 1 7
12 5 13
- Find the determinant of the following 4×4 matrix by carefully choosing the row or column to work with.
E=
4 5 0 2
− 1 −3 0 3
4 8 1 5
− 3 2 0 9
Answers:
- detC=
∣∣
∣∣−^412
1 − 3
∣∣
∣∣= 12 − 12 = 0
- detD=
∣∣
∣∣
∣∣
4 8 3
0 1 7
12 5 13
∣∣
∣∣
∣∣=^4 ·^13 +^8 ·^7 ·^12 +^0 −^36 −^5 ·^7 ·^4 −^0 =^548
- Notice that the third column is made up with zeros and a one. Choose this column to make up the coefficients
because then instead of having to evaluate the determinant of four individual 3×3 matrices, you only need to do
one.
∣∣
∣∣
∣∣
∣∣4 5 0 2
− 1 −3 0 3
4 8 1 5
− 3 2 0 9
∣∣
∣∣
∣∣
∣∣
= 0 ·
∣∣
∣∣
∣∣
− 1 −3 3
4 8 5
− 3 2 9
∣∣
∣∣
∣∣−^0 ·
∣∣
∣∣
∣∣
4 5 2
4 8 5
−3 2 9
∣∣
∣∣
∣∣+^1 ·
∣∣
∣∣
∣∣
4 5 2
− 1 −3 3
− 3 2 9
∣∣
∣∣
∣∣−^0 ·
∣∣
∣∣
∣∣
4 5 2
− 1 −3 3
4 8 5
∣∣
∣∣
∣∣
=
∣∣
∣∣
∣∣
4 5 2
− 1 −3 3
− 3 2 9
∣∣
∣∣
∣∣
= 4 ·(− 3 )· 9 + 5 · 3 ·(− 3 )+ 2 ·(− 1 )· 2 − 18 − 24 −(− 45 )
=− 154
Practice
Find the determinants of each of the following matrices.