# Properties of determinants

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##### Intros
###### Lessons
1. Properties of Determinants Overview:
2. 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
3. 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 $B$
4. Applications to Determinants
• If det $(A) \neq 0$, then columns are linearly independent
• If det $(A) = 0$, then columns are linear dependent
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##### Examples
###### Lessons
1. Calculating the Determinant
Compute det $A$ by row reduction to echelon form, where .
1. Find the determinants where ,
2. Property/Applications of Determinants
You are given that and .
1. Show that det $(A)=$ det $(A^T)$.
2. Show that det $(AB)=$ det $A \; \cdot$ det $B$.
3. Use determinants to decide if the set of vectors are linearly independent
1. Proving the Property of Determinants
Show that if a square matrix $A$ invertible, then
$\det (A^{-1})=\frac{1}{\det (A)}$
1. Show that if 2 rows of a square matrix $A$ are the same, then det $A=0$