Chapter 2 — Matrix operations and determinant in small systems

In the previous chapter we wrote linear systems as \(A\mathbf{x}=\mathbf{b}\). Now we detail the operations that support this notation: vector addition, matrix-vector product, matrix-matrix product, and a first determinant interpretation for \(2\times 2\) systems.

2.1 Basic operations with vectors

Consider

\[ \mathbf{u}=\begin{bmatrix}1\\2\end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix}-3\\4\end{bmatrix} \]

Addition:

\[ \mathbf{u}+\mathbf{v}= \begin{bmatrix}1+(-3)\\2+4\end{bmatrix}= \begin{bmatrix}-2\\6\end{bmatrix} \]

Linear combination with scalar \(\lambda=2\):

\[ 2\mathbf{u}-\mathbf{v}= \begin{bmatrix}2\\4\end{bmatrix}- \begin{bmatrix}-3\\4\end{bmatrix}= \begin{bmatrix}5\\0\end{bmatrix} \]

These operations appear naturally in elimination steps and solution checks.

2.2 Matrix-vector product as a compact system

If

\[ A=\begin{bmatrix}2 & 1\\1 & -1\end{bmatrix} \quad \text{and} \quad \mathbf{x}=\begin{bmatrix}x_1\\x_2\end{bmatrix}, \]

then

\[ A\mathbf{x}= \begin{bmatrix} 2x_1+x_2 \\ x_1-x_2 \end{bmatrix} \]

Each matrix row generates one linear equation, which explains why \(A\mathbf{x}=\mathbf{b}\) describes a full system.

Worked example

If \(\mathbf{x}=\begin{bmatrix}2\\1\end{bmatrix}\), then

\[ A\mathbf{x}= \begin{bmatrix} 2\cdot 2 + 1\cdot 1 \\ 1\cdot 2 + (-1)\cdot 1 \end{bmatrix} = \begin{bmatrix}5\\1\end{bmatrix} \]

So \(\mathbf{x}\) solves the system with right-hand side \(\mathbf{b}=\begin{bmatrix}5\\1\end{bmatrix}\).

2.3 Matrix-matrix product and composition

For compatible matrices,

\[ C=AB, \quad c_{ij}=\sum_k a_{ik}b_{kj} \]

In a \(2\times 2\) example,

\[ A=\begin{bmatrix}1 & 2\\0 & 1\end{bmatrix}, \quad B=\begin{bmatrix}3 & -1\\4 & 2\end{bmatrix} \]

\[ AB= \begin{bmatrix} 1\cdot 3 + 2\cdot 4 & 1\cdot (-1)+2\cdot 2 \\ 0\cdot 3 + 1\cdot 4 & 0\cdot (-1)+1\cdot 2 \end{bmatrix} = \begin{bmatrix}11 & 3\\4 & 2\end{bmatrix} \]

This product is central for computational pipelines of successive linear operations.

2.4 Determinant in \(2\times 2\): invertibility intuition

For

\[ A=\begin{bmatrix}a & b\\c & d\end{bmatrix}, \]

the determinant is

\[ \det(A)=ad-bc \]

For \(2\times 2\) systems, it gives a quick uniqueness test.

Situation	Condition on \(\det(A)\)	Consequence for \(A\mathbf{x}=\mathbf{b}\)
Invertible matrix	\(\det(A)\neq 0\)	Unique solution
Singular matrix	\(\det(A)=0\)	No unique solution (none or infinitely many)

Short example

Let

\[ A=\begin{bmatrix}2 & 1\\4 & 2\end{bmatrix} \]

Then

\[ \det(A)=2\cdot 2 - 1\cdot 4 = 0 \]

The rows are proportional, so the equations are not independent. Geometrically, this corresponds to parallel or coincident lines in the plane.

2.5 Connection back to Volume 2

In Volume 2, the crossing of two linear functions appeared as an intersection point. Here, the same problem is rewritten in matrix form.

If

\[ \begin{cases} y=2x+1 \\ y=-x+7 \end{cases} \Rightarrow \begin{cases} 2x-y=-1 \\ x+y=7 \end{cases} \]

then

\[ \underbrace{\begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}}_{A} \underbrace{\begin{bmatrix}x\\y\end{bmatrix}}_{\mathbf{x}} = \underbrace{\begin{bmatrix}-1\\7\end{bmatrix}}_{\mathbf{b}} \]

and

\[ \det(A)=2\cdot 1 - (-1)\cdot 1 = 3 \neq 0, \]

so the system has one unique solution, consistent with the unique intersection.

2.6 Chapter wrap-up

Linear algebra makes multi-variable models compact and operational. Vectors and matrices do not replace geometric intuition; they formalize it so it can scale to larger computational problems.