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8.4 A Determinant Formula for A −1 259

8.4 A Determinant Formula for A −1

When |A| = 0, A has an inverse. Furthermore, there is a formula for the elements of this inverse
in terms of determinants formed from elements of A.

THEOREM 8.4 Elements of a Matrix Inverse

Let A be a nonsingular n × n matrix and define an n × n matrix B =[b ij ] by
1
b ij = (−1) i+ j M ji .
|A|
Then B = A .
−1
Note that the i, j element of B is defined in terms of (−1) i+ j M ji , the cofactor of a ji (not a ij ).
We can see why this construction yields A −1 by explicitly multiplying the two matrices. By
the definition of matrix multiplication, the i, j element of AB is
n n
1 j+k
(AB) ij = a ik b kj = (−1) a ik M jk . (8.6)
|A|
k=1 k=1
Now consider two cases. If i = j the sum in equation (8.6) is exactly the cofactor expansion
of |A| by row i. The main diagonal elements of AB are therefore 1.
If i = j, the sum in equation (8.6) is the cofactor expansion by row j of the determinant of
the matrix formed from A by replacing row j by row i. But this matrix has two identical rows,
so its determinant is zero and the off-diagonal elements of AB are all zero. This means that
AB = I n .
Similarly, BA = I n .

EXAMPLE 8.6
Let
⎛ ⎞
−24 1
A = ⎝ 6 3 −3 ⎠ .
2 9 −5
It is routine to compute |A|= 120 so A is nonsingular. We will determine A −1 by computing the
elements of the matrix B of Theorem 8.4:

1 1 3 −3 12 1
b 11 = M 11 = = = ,
120 120 9 −5 120 10

1 1 4 1 29
b 12 = (−1)M 21 =− = ,
120 120 9 −5 120

1 1 4 1 1
b 13 = M 31 = =− ,
120 120 3 −3 8

1 1 6 −3 1
b 21 =− M 12 =− = ,
120 120 2 −5 5

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October 14, 2010 14:26 THM/NEIL Page-259 27410_08_ch08_p247-266