Given an n×n matrix A, by Ai(b) we denote the matrix obtained by replacing the ith column of A with b.
Cramer’s Rule: Let A be an n×n invertible matrix. For any b in Rn, the solution to Ax=b is given by xi=detAdetAi(b).
This is not very useful for calculations, but makes some theoretical proofs simpler.
Proof. Consider Ii(x).
Ii(x)detIi(x)AIi(x)detAi(b)xi=⎝⎜⎜⎜⎜⎜⎜⎛10...0...00100............x1x2xixn............0001⎠⎟⎟⎟⎟⎟⎟⎞=∣∣∣∣∣∣∣∣∣∣∣∣10...0...00100............x1x2xixn............0001∣∣∣∣∣∣∣∣∣∣∣∣=xi∣∣∣∣∣∣∣∣∣∣∣∣10...0...00100............x1x21xn............0001∣∣∣∣∣∣∣∣∣∣∣∣=xi∣∣∣∣∣∣∣∣∣∣∣∣10...0...00100............0010............0001∣∣∣∣∣∣∣∣∣∣∣∣=xi detIn=xi=A[e1e2...xei+1...en]=[Ae1Ae2...AxAei+1...Aen]=[Ae1Ae2...bAei+1...Aen]=Ai(b)=detAIi(x)=detA detIi(x)=xidetA=detAdetAi(b).
Ex. Let s be a parameter
{3sx1−2x2=4−6x1+sx2=1
and assume that the system has a unique solution.
AdetAsA1(b)detA1(b)A2(b)detA2(b)=(3s−6−2s)=3s2−12=3(s2−4)=3(s−2)(s+2)=2,−2=(41−2s)=4s+2=(3s−641)=3s+24
By Cramer’s Rule:
x1x2=3(s−2)(s+2)4s+2=3(s−2)(s+2)3s+24
Cramer’s Rule also provides a formula to find the inverse of matrices. (You really shouldn’t use this for computations because it’s harder than other methods we’ve learned.)
A−1=detA1⎝⎜⎜⎛C11C12...C1nC21C22C2n.........Cn1Cn2Cnn⎠⎟⎟⎞
where Cij are cofactors that were defined previously. (The matrix on the right here is called the classical adjoint of A.)
If A is a 2×2 matrix, the area of the parallelogram determined by the columns of A is ∣detA∣. This also applies to the volume of parallelopipeds for 3×3 matrices, and for higher-dimensional equivalents of parallelograms and volume for larger square matrices.
By determined by the columns of A, we mean the parallelogram that has (0,0) and each column of A as vertices. The remaning vertices can be determined from this information.
Let T:R2→R2 be a linear transformation determined by the matrix A. If S is an object in R2 with finite area, then the area of T(S) is the product of ∣detA∣ and the area of S. (T(S) means applying T to every point in S.)