Foundations — Properties of determinants

This page is the ground floor. The parent note fires off symbols like $a_{ij}$, $\sigma$, $\mathrm{sgn}$, $R_i$, $\det A$, $A^{\top }$ — and quietly assumes you already picture what each one means. Here we build every one of them from nothing, in the order that each depends on the last.

0. What a "matrix" even is

Look at the figure: the matrix is a grid. The horizontal strip highlighted in blue is a row; the vertical strip in orange is a column. Nothing more mysterious than a seating chart.

We call a matrix square when it has the same number of rows as columns ($2\times 2$, $3\times 3$, ...). Determinants are only defined for square matrices — you need equal rows and columns for the "area" idea to make sense, and we'll see exactly why in section 5.

1. The entry symbol $a_{ij}$

In the figure below, we point an arrow at $a_{23}$: walk down to row 2, then across to column 3. The topic needs this because the determinant formula picks entries one at a time by address, and if you read the address backwards every property comes out wrong.

2. The row symbol $R_i$ (and why the topic loves rows)

The parent note writes the determinant as a function $D(R_1,R_2,\dots ,R_n)$. Read that as: "feed in the rows one bundle at a time, get a number out." Why bundle rows? Because every property (swap rows, scale a row, add rows) is a story about moving these strips around — and it's far easier to picture three strips than nine loose numbers.

3. The determinant symbol: $\det A$ and the vertical bars $|\cdot |$

For a $2\times 2$ matrix the recipe is: $\det \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc.$

What does $ad-bc$ look like? It is exactly the area of the parallelogram made by the two row-arrows $(a,b)$ and $(c,d)$ — with a sign for orientation. The figure builds it: the big rectangle has area $(a+?)$... more simply, the shaded blue region is the parallelogram, and $ad-bc$ is its signed area.

This is the foundation of Determinant — Definition and Expansion by Minors and Area of a Triangle using Determinants.

4. Summation $\sum$ — a machine that adds a list

You don't need to fear ${\sum }_{\sigma }\mathrm{sgn}(\sigma )a_{1\sigma (1)}\cdots$ yet. It just says: make a big sum, where each piece grabs one entry per row and per column, and stick a $\pm$ on it. The next two sections define $\sigma$ and $\mathrm{sgn}$ so that sum stops looking like hieroglyphics.

5. Permutation $\sigma$ — a way to rearrange, pictured as arrows

The figure shows two shuffles for a $3\times 3$: the identity (every arrow goes straight across — this picks the diagonal) and a swap (two arrows cross). Each set of non-crossing/crossing arrows is one $\sigma$, and it selects exactly one entry from every row and column.

Why does the topic need permutations? Because the determinant is the only natural way to combine "one entry per row and column" — and permutations are the exhaustive list of ways to do that.

6. The sign $\mathrm{sgn}(\sigma )$ — even or odd number of swaps

This underpins P2 and P3 on the parent page.

7. Transpose $A^{\top }$ — flip across the diagonal

Picture a mirror line running down the diagonal; every entry hops to its reflection. The parent's P1 says $\det (A^{\top })=\det A$ — meaning "whatever is true for rows is equally true for columns." That's why the note can prove things about rows and get columns for free.

8. Product $AB$ and inverse $A^{-1}$ (just enough to read P9)

Prerequisite map

Equipment checklist

Connections

• Properties of determinants
• Determinant — Definition and Expansion by Minors
• Cofactors and Adjoint of a Matrix
• Inverse of a Matrix via Adjoint
• Cramer's Rule and Systems of Linear Equations
• Elementary Row Operations & Rank
• Area of a Triangle using Determinants
• Linear Transformations & Scaling of Area