In this section we specialize and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D:Determinants, Chapter E:Eigenvalues Chapter LT:Linear Transformations, Chapter R:Representations) that these matrices are especially important.

## Nonsingular Matrices

Our theorems will now establish connections between systems of equations (homogeneous or otherwise), augmented matrices representing those systems, coefficient matrices, constant vectors, the reduced row-echelon form of matrices (augmented and coefficient) and solution sets. Be very careful in your reading, writing and speaking about systems of equations, matrices and sets of vectors. A system of equations is not a matrix, a matrix is not a solution set, and a solution set is not a system of equations. Now would be a great time to review the discussion about speaking and writing mathematics in technique L.

Definition SQM (Square Matrix) A matrix with $m$ rows and $n$ columns is square if $m=n$. In this case, we say the matrix has size $n$. To emphasize the situation when a matrix is not square, we will call it rectangular.

We can now present one of the central definitions of linear algebra.

Definition NM (Nonsingular Matrix) Suppose $A$ is a square matrix. Suppose further that the solution set to the homogeneous linear system of equations $\linearsystem{A}{\zerovector}$ is $\set{\zerovector}$, i.e. the system has only the trivial solution. Then we say that $A$ is a nonsingular matrix. Otherwise we say $A$ is a singular matrix.

We can investigate whether any square matrix is nonsingular or not, no matter if the matrix is derived somehow from a system of equations or if it is simply a matrix. The definition says that to perform this investigation we must construct a very specific system of equations (homogeneous, with the matrix as the coefficient matrix) and look at its solution set. We will have theorems in this section that connect nonsingular matrices with systems of equations, creating more opportunities for confusion. Convince yourself now of two observations, (1) we can decide nonsingularity for any square matrix, and (2) the determination of nonsingularity involves the solution set for a certain homogeneous system of equations.

Notice that it makes no sense to call a system of equations nonsingular (the term does not apply to a system of equations), nor does it make any sense to call a $5\times 7$ matrix singular (the matrix is not square).

Example S: A singular matrix, Archetype A.

Example NM: A nonsingular matrix, Archetype B.

Notice that we will not discuss Example HISAD as being a singular or nonsingular coefficient matrix since the matrix is not square.

The next theorem combines with our main computational technique (row-reducing a matrix) to make it easy to recognize a nonsingular matrix. But first a definition.

Definition IM (Identity Matrix) The $m\times m$ identity matrix, $I_m$, is defined by

\begin{align*} \matrixentry{I_m}{ij}&= \begin{cases} 1 & i=j\\ 0 & i\neq j \end{cases}        1\leq i,\,j\leq m \end{align*}

(This definition contains Notation IM.)

Example IM: An identity matrix.

Notice that an identity matrix is square, and in reduced row-echelon form. So in particular, if we were to arrive at the identity matrix while bringing a matrix to reduced row-echelon form, then it would have all of the diagonal entries circled as leading 1's.

Theorem NMRRI (Nonsingular Matrices Row Reduce to the Identity matrix) Suppose that $A$ is a square matrix and $B$ is a row-equivalent matrix in reduced row-echelon form. Then $A$ is nonsingular if and only if $B$ is the identity matrix.

Notice that since this theorem is an equivalence it will always allow us to determine if a matrix is either nonsingular or singular. Here are two examples of this, continuing our study of Archetype A and Archetype B.

Example SRR: Singular matrix, row-reduced.

Example NSR: Nonsingular matrix, row-reduced.

## Null Space of a Nonsingular Matrix

Nonsingular matrices and their null spaces are intimately related, as the next two examples illustrate.

Example NSS: Null space of a singular matrix.

Example NSNM: Null space of a nonsingular matrix.

These two examples illustrate the next theorem, which is another equivalence.

Theorem NMTNS (Nonsingular Matrices have Trivial Null Spaces) Suppose that $A$ is a square matrix. Then $A$ is nonsingular if and only if the null space of $A$, $\nsp{A}$, contains only the zero vector, i.e. $\nsp{A}=\set{\zerovector}$.

The next theorem pulls a lot of big ideas together. Theorem NMUS tells us that we can learn much about solutions to a system of linear equations with a square coefficient matrix by just examining a similar homogeneous system.

Theorem NMUS (Nonsingular Matrices and Unique Solutions) Suppose that $A$ is a square matrix. $A$ is a nonsingular matrix if and only if the system $\linearsystem{A}{\vect{b}}$ has a unique solution for every choice of the constant vector $\vect{b}$.

This theorem helps to explain part of our interest in nonsingular matrices. If a matrix is nonsingular, then no matter what vector of constants we pair it with, using the matrix as the coefficient matrix will always yield a linear system of equations with a solution, and the solution is unique. To determine if a matrix has this property (non-singularity) it is enough to just solve one linear system, the homogeneous system with the matrix as coefficient matrix and the zero vector as the vector of constants (or any other vector of constants, see exercise MM.T10).

Formulating the negation of the second part of this theorem is a good exercise. A singular matrix has the property that for some value of the vector $\vect{b}$, the system $\linearsystem{A}{\vect{b}}$ does not have a unique solution (which means that it has no solution or infinitely many solutions). We will be able to say more about this case later (see the discussion following Theorem PSPHS). Square matrices that are nonsingular have a long list of interesting properties, which we will start to catalog in the following, recurring, theorem. Of course, singular matrices will then have all of the opposite properties. The following theorem is a list of equivalences.

Theorem NME1 (Nonsingular Matrix Equivalences, Round 1) Suppose that $A$ is a square matrix. The following are equivalent.

1. $A$ is nonsingular.
2. $A$ row-reduces to the identity matrix.
3. The null space of $A$ contains only the zero vector, $\nsp{A}=\set{\zerovector}$.
4. The linear system $\linearsystem{A}{\vect{b}}$ has a unique solution for every possible choice of $\vect{b}$.

Finally, you may have wondered why we refer to a matrix as nonsingular when it creates systems of equations with single solutions (Theorem NMUS)! I've wondered the same thing. We'll have an opportunity to address this when we get to Theorem SMZD. Can you wait that long?