The theory of matrices and determinants 239
7. Evaluate
3∠60 ◦
j 2
1
0
(1 + j ) 2∠30 ◦
0
2
j 5
26.94∠−139.52 ◦ or
(−20.49 − j 17.49)
8. Find the eigenvalues λ that satisfy the following equations:
(a)
(2 − λ)
2
−1
(5 − λ)
= 0
(b)
(5 − λ)
7
−5
0
(4 − λ)
−1
2
8
(−3 − λ)
= 0
(You may need to refer to chapter 1, pages
8–12, for the solution of cubic equations).
[(a) λ =3 or 4 (b) λ =1 or 2 or 3]
22.7 The inverse or reciprocal of a
3 by 3 matrix
The adjoint of a matrix A is obtained by:
(i) forming a matrix B of the cofactors of A, and
(ii) transposing matrix B to give B T , where B T is the
matrix obtained by writing the rows of B as the
columns of B T . Then adj A = B T .
The inverse of matrix A, A −1 is given by
A
−1
=
adj A
|A|
where adj A is the adjoint of matrix A and |A| is the
determinant of matrix A.
Problem 17. Determine the inverse of the matrix
⎛
⎜
⎝
3
4 −1
2
0
7
1 −3 −2
⎞
⎟
⎠
The inverse of matrix A, A −1 =
adj A
|A|
The adjoint of A is found by:
(i) obtaining the matrix of the cofactors of the elements, and
(ii) transposing this matrix.
The cofactor of element 3 is +
0
7
−3 −2
= 21.
The cofactor of element 4 is −
2
7
1 −2
= 11, and so on.
The matrix of cofactors is
⎛
⎝
21
11 −6
11 −5 13
28 −23 −8
⎞
⎠
The transpose of the matrix of cofactors, i.e. the adjoint
of the matrix, is obtained by writing the rows as columns,
and is
⎛
⎝
21 11
28
11 −5 −23
−6 13 −8
⎞
⎠
From Problem 14, the determinant of
3
4 −1
2
0
7
1 −3 −2
is 113.
Hence the inverse of
⎛
⎝
3
4 −1
2
0
7
1 −3 −2
⎞
⎠ is
⎛
⎝
21 11
28
11 −5 −23
−6 13 −8
⎞
⎠
113
or
1
113
⎛
⎝
21 11
28
11 −5 −23
−6 13 −8
⎞
⎠
Problem 18. Find the inverse of
⎛
⎜
⎝
1
5 −2
3 −1
4
−3
6 −7
⎞
⎟
⎠
Inverse =
adjoint
determinant
The matrix of cofactors is
⎛
⎝
−17
9
15
23 −13 −21
18 −10 −16
⎞
⎠
The transpose of the matrix of cofactors (i.e. the
adjoint) is
⎛
⎝
−17
23
18
9 −13 −10
15 −21 −16
⎞
⎠