# Properties of determinants

### Properties of determinants

#### Lessons

Let $A$ be a $n \times n$ square matrix. Then
1) If a multiple of one row of matrix $A$ is added to another row to produce matrix $B$, then det $B=$ det $A$
2) If two rows of $A$ are interchanged to produce matrix $B$, then det $B=-$ det $A$
3) If one row of $A$ is multiplied by $k$ to produce matrix $B$, then det $B=k \; \cdot$det $A$

If $A$ is an $n \times n$ matrix, then det$(A^T )=$ det $A$

A square matrix $A$ is invertible if and only if det $A \neq 0$.

The Multiplicative Property
If $A$ and $B$ are $n \times n$ matrices, then det $AB=$ (det $A$)(det $B$)

It may be useful to know that the determinant of a triangular matrix is the product of the diagonal entries. For example,
• Introduction
Properties of Determinants Overview:
a)
Row Operation Property
• Adding/Subtracting Rows → det $B=$ det $A$
• Interchanging rows → det $B=-$ det $A$
• Multiplying rows → det $B=k\; \cdot$ det $A$
• Triangular Matrices

b)
The Multiplicative Property/Other Properties
• det $(A)=$ det $(A^T)$
$A$ is invertible if and only if det $A \neq 0$
• det $(AB)=$ det $A \; \cdot$ det $B$

c)
Applications to Determinants
• If det $(A) \neq 0$, then columns are linearly independent
• If det $(A) = 0$, then columns are linear dependent

• 1.
Calculating the Determinant
Compute det $A$ by row reduction to echelon form, where .

• 2.
Find the determinants where ,
a)

b)

• 3.
Property/Applications of Determinants
You are given that and .
a)
Show that det $(A)=$ det $(A^T)$.

b)
Show that det $(AB)=$ det $A \; \cdot$ det $B$.

• 4.
Use determinants to decide if the set of vectors are linearly independent

• 5.
Proving the Property of Determinants
Show that if a square matrix $A$ invertible, then
$\det (A^{-1})=\frac{1}{\det (A)}$

• 6.
Show that if 2 rows of a square matrix $A$ are the same, then det $A=0$