Determinants

Notes

## 1. Determinant

A determinant is a numerical value associated with a **square matrix**.

If **A** is a square matrix, its determinant is represented by:

\[
|A|
\]

or

\[
\det(A)
\]

**Important:** Determinants are defined only for square matrices.

---

## 2. Determinant of Order 1

For a matrix containing only one element:

\[
A=
\begin{bmatrix}
a
\end{bmatrix}
\]

its determinant is:

\[
|A|=a
\]

---

## 3. Determinant of Order 2

Let:

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

Then:

\[
|A|=
\begin{vmatrix}
a & b\\
c & d
\end{vmatrix}
\]

and its value is:

\[
|A|=ad-bc
\]

### Example

Evaluate:

\[
\begin{vmatrix}
2 & 3\\
4 & 5
\end{vmatrix}
\]

Using:

\[
\begin{vmatrix}
a & b\\
c & d
\end{vmatrix}
=ad-bc
\]

we get:

\[
=2(5)-3(4)
\]

\[
=10-12
\]

\[
=-2
\]

Therefore:

\[
\boxed{-2}
\]

---

## 4. Determinant of Order 3

Consider:

\[
A=
\begin{bmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\]

Its determinant is written as:

\[
|A|=
\begin{vmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{vmatrix}
\]

A determinant can be expanded along **any row or any column**.

### Expansion Along First Row

\[
|A|
=
a_{11}
\begin{vmatrix}
a_{22} & a_{23}\\
a_{32} & a_{33}
\end{vmatrix}
-
a_{12}
\begin{vmatrix}
a_{21} & a_{23}\\
a_{31} & a_{33}
\end{vmatrix}
+
a_{13}
\begin{vmatrix}
a_{21} & a_{22}\\
a_{31} & a_{32}
\end{vmatrix}
\]

### Sign Pattern

The sign pattern used while finding cofactors is:

\[
\begin{bmatrix}
+ & - & +\\
- & + & -\\
+ & - & +
\end{bmatrix}
\]

---

# Properties of Determinants

Properties of determinants are very useful for simplifying complicated determinants.

## 5. Property 1 – Transpose Property

The value of a determinant remains unchanged if its rows and columns are interchanged.

Therefore:

\[
|A|=|A^T|
\]

For example:

\[
\begin{vmatrix}
a & b\\
c & d
\end{vmatrix}
=
\begin{vmatrix}
a & c\\
b & d
\end{vmatrix}
\]

---

## 6. Property 2 – Interchange of Two Rows or Columns

If any two rows or any two columns are interchanged, the sign of the determinant changes.

If:

\[
R_i \leftrightarrow R_j
\]