Section 10.5 Theory
We have discussed the reasoning behind many of the below facts in Section 10.3, so we will omit some of the formal proofs.
Subsection 10.5.1 Adjoints and inverses
First, we record the adjoint inversion formula we have discovered.
Theorem 10.5.1. Inversion by adjoint.
Remark 10.5.2.
Based on our computations for the case in Subsection 10.4.1, if is then the statement of the theorem above is exactly the same as Proposition 5.5.4.
Subsection 10.5.2 Determinants determine invertibility
As we saw in Subsection 10.3.2, there is a stronger connection between the determinant and invertibility, which we now state here more formally by adding a new statement to Theorem 6.5.2.
Theorem 10.5.3. Characterizations of invertibility.
For a square matrix the following are equivalent.
- Matrix
is invertible. - Every linear system that has
as a coefficient matrix has one unique solution. - The homogeneous system
has only the trivial solution. - There is some linear system that has
as a coefficient matrix and has one unique solution. - The rank of
is equal to the size of - The RREF of
is the identity. - Matrix
can be expressed as a product of some number of elementary matrices. - The determinant of
is nonzero.
In particular, a square matrix is invertible if and only if its determinant is nonzero.
Remark 10.5.4.
In the last sentence of the theorem, the connecting phrase “if and only if” between the two conditions is just a different way to say that the two conditions are equivalent. And recall that conditions are equivalent when they have to be either all true or all false at the same time. Rephrasing in terms of the “all false” scenario, we could also say that a square matrix is singular if and only if its determinant is zero.
Subsection 10.5.3 Determinant formulas
Here we collect the determinant formulas from Subsection 10.3.3. First we look at a special case, previously considered in Discovery 10.4 and Subsubsection 10.3.3.1, of the multiplicative formula for determinants.
Lemma 10.5.5. Determinant is multiplicative: elementary case.
Proof.
There are three cases to consider here, based on the type of elementary matrix we are dealing with.
Case represents swapping rows.
This establishes (✶) in this case.
Case represents multiplying a row by .
This establishes (✶) in this case.
Case represents adding a multiple of one row to another.
The product represents the result of adding a multiple of a row to another in so is equal to But also (Part 3 of Proposition 9.4.5), so
establishing (✶) in this case.
With the above lemma established, we can consider the general multiplicative formula for determinants.
Proposition 10.5.6. Determinant is multiplicative: general case.
A determinant of a product of square matrices is the product of the determinants of those matrices. In particular, the following hold.
- If
and are square matrices of the same size, then - If
are square matrices of the same size, then
Proof outline for Statement 1.
There are two cases to consider.
Case is invertible.
In this case, can be expressed as a product of elementary matrices (Theorem 10.5.3), and so Lemma 10.5.5 can be repeatedly applied to obtain the desired equality.
In Discovery 10.5 and Subsubsection 10.3.3.2, we worked under the assumption that could be expressed as a product of three elementary matrices, but the calculations and logic used there would work no matter how many elementary matrices were required in a product expression for
Case is singular.
In this case, by our newly added statement in the list of Theorem 10.5.3, so we have
as well. But we also know that if is singular, then the product must also be singular (Statement 1 of Proposition 6.5.8). So again we can apply the equivalence of Statement 1 and Statement 8 of Theorem 10.5.3 to obtain
Since both LHS and RHS are equal to they are equal to each other.
Proof outline for Statement 2.
This result can be obtained by repeated applications of the formula in Statement 1, one at a time.
Remark 10.5.7.
Lemma 10.5.5 and the proof of Proposition 10.5.6 connect to Proposition 9.4.2 (which includes the formula as one of its statements) by the fact that an scalar matrix is the product of elementary matrices, one for each of the operations multiply row by .
Proposition 10.5.8. Determinant of an inverse.
The determinant of an inverse is the inverse of the determinant. That is, if is an invertible matrix then
Subsection 10.5.4 Cramer’s rule
Finally, we formally record Cramer’s rule (discussed in Subsection 10.3.5).
Theorem 10.5.9. Cramer’s rule.
If system has invertible square coefficient matrix then the value of variable in the one unique solution to the system is