The theory of matrices and determinants 239
- Evaluate
∣ ∣ ∣ ∣ ∣ ∣
3 ∠ 60 ◦ j 21
0 ( 1 +j) 2 ∠ 30 ◦
02 j 5∣ ∣ ∣ ∣ ∣ ∣ [26. 94 ∠− 139. 52 ◦or
(− 20. 49 −j 17. 49 )]- Find the eigenvaluesλthat satisfy the follow-
ing equations:
(a)∣
∣
∣
∣( 2 −λ) 2
− 1 ( 5 −λ)∣
∣
∣
∣=^0(b)∣ ∣ ∣ ∣ ∣ ∣
( 5 −λ) 7 − 5
0 ( 4 −λ) − 1
28 (− 3 −λ)∣ ∣ ∣ ∣ ∣ ∣= 0(You may need to refer to chapter 1, pages
8–12, for the solution of cubic equations).
[(a)λ=3or4 (b)λ=1 or 2 or 3]22.7 The inverse or reciprocal of a 3 by 3 matrix
Theadjointof a matrixAis obtained by:
(i) forming a matrixBof the cofactors ofA,and
(ii) transposingmatrixBto giveBT,whereBTis the
matrix obtained by writing the rows ofBas the
columns ofBT.ThenadjA=BT.Theinverse of matrixA,A−^1 is given by
A−^1 =adjA
|A|where adjAis the adjoint of matrixAand|A|is the
determinant of matrixA.
Problem 17. Determine the inverse of the matrix
⎛
⎜
⎝34 − 1
207
1 − 3 − 2⎞
⎟
⎠The inverse of matrixA,A−^1 =
adjA
|A|The adjoint ofAis found by:(i) obtaining the matrix of the cofactors of the ele-
ments, and(ii) transposing this matrix.The cofactor of element 3 is+∣
∣
∣
∣07
− 3 − 2∣
∣
∣
∣=21.The cofactor of element 4 is−∣
∣
∣
∣27
1 − 2∣
∣
∣
∣=11, and so on.The matrix of cofactors is⎛
⎝21 11 − 6
11 − 513
28 − 23 − 8⎞
⎠The transpose of the matrix of cofactors, i.e. the adjoint
ofthematrix,isobtainedbywritingtherowsascolumns,and is⎛
⎝21 11 28
11 − 5 − 23
− 613 − 8⎞
⎠From Problem 14, the determinant of∣ ∣ ∣ ∣ ∣ ∣
34 − 1
207
1 − 3 − 2∣ ∣ ∣ ∣ ∣ ∣is 113.Hence the inverse of⎛
⎝34 − 1
207
1 − 3 − 2⎞
⎠is⎛
⎝21 11 28
11 − 5 − 23
− 613 − 8⎞
⎠113or
1
113⎛
⎝21 11 28
11 − 5 − 23
− 613 − 8⎞
⎠Problem 18. Find the inverse of
⎛
⎜
⎝15 − 2
3 − 14
− 36 − 7⎞
⎟
⎠Inverse=adjoint
determinantThe matrix of cofactors is⎛
⎝− 17 9 15
23 − 13 − 21
18 − 10 − 16⎞
⎠The transpose of the matrix of cofactors (i.e. theadjoint) is⎛
⎝−17 23 18
9 − 13 − 10
15 − 21 − 16⎞
⎠