Matrix Inverses and Nonsingular Matrices

We saw in that if a square matrix $A$ is nonsingular, then there is a matrix $B$ so that $AB=I_n$. In other words, $B$ is halfway to being an inverse of $A$. We will see in this section that $B$ automatically fulfills the second condition ($BA=I_n$). showed us that the coefficient matrix from had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

1. 4x_1 + 10x_2 &= 12\\ 2x_1 + 6x_2 &= 4
2. \begin{bmatrix} 2 & 3 & 1\\ 1 & -2 & -3\\ -2 & 4 & 6 \end{bmatrix}
3. A=\begin{bmatrix} \frac{1}{\sqrt{22}}\left(4+2i\right) & \frac{1}{\sqrt{374}}\left(5+3i\right) \\ \frac{1}{\sqrt{22}}\left(-1-i\right) & \frac{1}{\sqrt{374}}\left(12+14i\right) \\ \end{bmatrix}

We need a couple of technical results for starters. Some books would call these minor, but essential, results lemmas. We'll just call 'em theorems. See for more on the distinction.

The first of these technical results is interesting in that the hypothesis says something about a product of two square matrices and the conclusion then says the same thing about each individual matrix in the product. This result has an analogy in the algebra of complex numbers: suppose $\alpha,\,\beta\in\complexes$, then $\alpha\beta\neq 0$ if and only if $\alpha\neq 0$ and $\beta\neq 0$. We can view this result as suggesting that the term nonsingular for matrices is like the term nonzero for scalars. Consider too that we know singular matrices, as coefficient matrices for systems of equations, will sometimes lead to systems with no solutions, or systems with infinitely many solutions (). What do linear equations with zero look like? Consider $0x=5$, which has no solution, and $0x=0$, which has infinitely many solutions. In the algebra of scalars, zero is exceptional (meaning different, not better), and in the algebra of matrices, singular matrices are also the exception. While there is only one zero scalar, and there are infinitely many singular matrices, we will see that singular matrices are a distinct minority.

This is a powerful result in the forward direction, because it allows us to begin with a hypothesis that something complicated (the matrix product $AB$) has the property of being nonsingular, and we can then conclude that the simpler constituents ($A$ and $B$ individually) then also have the property of being nonsingular. If we had thought that the matrix product was an artificial construction, results like this would make us begin to think twice.

The contrapositive of this entire result is equally interesting. It says that $A$ or $B$ (or both) is a singular matrix if and only if the product $AB$ is singular. (See .)

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Recall that the adjoint of a matrix is $\adjoint{A}=\transpose{\left(\conjugate{A}\right)}$ ().

This condition may seem rather far-fetched at first glance. Would there be any matrix that behaved this way? Well, yes, here is one.

Unitary matrices do not have to look quite so gruesome. Here is a larger one that is a bit more pleasing.

If a matrix $A$ has only real number entries (we say it is a real matrix) then the defining property of being unitary simplifies to $\transpose{A}A=I_n$. In this case we, and everybody else, call the matrix orthogonal, so you may often encounter this term in your other reading when the complex numbers are not under consideration.

Unitary matrices have easily computed inverses. They also have columns that form orthonormal sets. Here are the theorems that show us that unitary matrices are not as strange as they might initially appear.

When using vectors and matrices that only have real number entries, orthogonal matrices are those matrices with inverses that equal their transpose. Similarly, the inner product is the familiar dot product. Keep this special case in mind as you read the next theorem.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.