2.7 Inverse Matrix
1. If A is a square matrix, B is another square matrix and A × B = B × A = I, then matrix A is the inverse matrix of matrix B and vice versa. Matrix A is called the inverse matrix of B for multiplication and vice versa.
2. The symbol A-1 denotes the inverse matrix of A.
3. Inverse matrices can only exist for square matrices but not all square matrices have inverse matrices.
4. If AB ≠ I or BA ≠ I, then A is not the inverse of B and B is not the inverse of A.
Example 1:
Determine whether matrix
A=(2915)A=(2915)
is an inverse matrix of matrix
B=(5−9−12).B=(5−9−12).
Solution:
AB=(2915)(5−9−12)=(2×5+9×−12×−9+9×21×5+5×−11×−9+5×2)=(10+(−9)−18+185+(−5)−9+10)=(1001)=IAB=(5−9−12)(2915)=(5×2+(−9)×15×9+(−9)×5−1×2+2×1−1×9+2×5)=(10+(−9)18−18−2+2−9+10)=(1001)=IAB=BA=I∴A is the inverse matrix of B and vice versa.AB=(2915)(5−9−12)=(2×5+9×−12×−9+9×21×5+5×−11×−9+5×2)=(10+(−9)−18+185+(−5)−9+10)=(1001)=IAB=(5−9−12)(2915)=(5×2+(−9)×15×9+(−9)×5−1×2+2×1−1×9+2×5)=(10+(−9)18−18−2+2−9+10)=(1001)=IAB=BA=I∴A is the inverse matrix of B and vice versa.
5. The inverse of a matrix may also be found using a formula.
If
A=(abcd)A=(abcd)
, then the inverse matrix of A, A-1, is given by the formula below.
A−1=1ad−bc(d−b−ca), where ad−bc≠0
A−1=1ad−bc(d−b−ca), where ad−bc≠0
6. ad – bc is known as the determinant of matrix A.
7. If the determinant, ad – bc = 0, then the inverse matrix of A does not exist.
Example 2:
Find the inverse matrix of
A=(61−9−1)
using the formula.Solution:
A=(61−9−1)a=6,b=1,c=−9,d=−1A−1=1ad−bc(d−b−ca)A−1=16×−1−(1×−9)(−1−196)A−1=1−6+9(−1−196)A−1=13(−1−196)=(−13−1332)
Example 3:
The inverse matrix of
(−72−92)isr(2s9t).
Find the value of r, of s and of t.Solution:
LetA=(−72−92)A−1=1−7×2−(−9)×2(2−29−7)A−1=14(2−29−7)∴r(2s9t)=14(2−29−7)By comparison,r=14,s=−2,t=−7.