A First Course in Linear Algebra, 2017a

134 Determinants
The cofactor matrix is
and so the inverse is
⎡
1
⎣
e t
⎡
C (t)= ⎣
1 0 0
0 e t cost e t sint
0 −e t sint e t cost
1 0 0
0 e t cost e t sint
0 −e t sint e t cost
⎤
⎦
T
⎡
= ⎣
⎤
⎦
e −t 0 0
0 cost −sint
0 sint cost
⎤
⎦
♠
3.2.2 Cramer’s Rule
Another context in which the formula given in Theorem 3.40 is important is Cramer’s Rule. Recall that
we can represent a system of linear equations in the form AX = B, where the solutions to this system
are given by X. Cramer’s Rule gives a formula for the solutions X in the special case that A is a square
invertible matrix. Note this rule does not apply if you have a system of equations in which there is a
different number of equations than variables (in other words, when A is not square), or when A is not
invertible.
Suppose we have a system of equations given by AX = B, and we want to find solutions X which
satisfy this system. Then recall that if A −1 exists,
AX = B
A −1 (AX) = A −1 B
(
A −1 A ) X = A −1 B
IX = A −1 B
X = A −1 B
Hence, the solutions X to the system are given by X = A −1 B. Since we assume that A −1 exists, we can use
the formula for A −1 given above. Substituting this formula into the equation for X,wehave
X = A −1 B = 1
det(A) adj(A)B
Let x i be the i th entry of X and b j be the j th entry of B. Then this equation becomes
x i =
n [ ] −1
∑ aij b j =
j=1
n
∑
j=1
1
det(A) adj(A) ij b j
where adj(A) ij is the ij th entry of adj(A).
By the formula for the expansion of a determinant along a column,
⎡
⎤
x i = 1 ∗ ··· b 1 ··· ∗
det(A) det ⎢
⎥
⎣ . . . ⎦
∗ ··· b n ··· ∗