Determinants & Inverses

Chapter 3 • Section 3-2

"When determinants meet inverses, magic happens! ✨"

🤝 The Product Theorem

Determinants play nicely with matrix multiplication:

$$ \det(AB) = \det A \cdot \det B $$

Inverse

Since $A A^{-1} = I$ and $\det I = 1$:

$$ \det(A^{-1}) = \frac{1}{\det A} $$

Only if $\det A \neq 0$!

Transpose

Flipping rows and columns doesn't change the volume:

$$ \det(A^T) = \det A $$

Example: When is it Invertible?

Find $c$ so $A = \begin{bmatrix} c & 1 & 0 \\ 0 & 2 & c \\ -1 & c & 5 \end{bmatrix}$ is invertible.

Calculate $\det A$ and set $\neq 0$:

$$ \det A = c(10-c^2) - (-1)(c) = 10c - c^3 + c = c(11-c^2) $$

So $A$ is invertible if $c \neq 0$ and $c \neq \pm\sqrt{11}$.

🤔 Quick Check Win $10

If $\det A = 2$ and $\det B = 3$, what is $\det(A^2 B^{-1})$?

Cramer's Rule

A fancy formula for solving $A\vec{x} = \vec{b}$ using determinants.

$$ x_i = \frac{\det A_i(\vec{b})}{\det A} $$

Where $A_i(\vec{b})$ is matrix $A$ with column $i$ replaced by $\vec{b}$.

Interactive Solver (2x2)

x₁ x₂

Special Matrices

The Adjugate

The transpose of the cofactor matrix.

$$ A^{-1} = \frac{1}{\det A} \adj(A) $$

A magic formula for the inverse!

Vandermonde

Used in polynomial interpolation.

$$ \det V = \prod_{i>j} (x_i - x_j) $$

Non-zero if all $x_i$ are distinct.

Example: Polynomial Interpolation

Find a polynomial $p(x) = r_0 + r_1 x + r_2 x^2$ passing through $(0,1), (1,2), (2,5)$.

Set up the system $V\vec{r} = \vec{y}$ where $V$ is Vandermonde:

$$ \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{bmatrix} \begin{bmatrix} r_0 \\ r_1 \\ r_2 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix} $$

Solving gives $r_0=1, r_1=0, r_2=1$. So $p(x) = 1 + x^2$.

🧠 Logic Check Win $20

If $A$ is an orthogonal matrix ($A^T = A^{-1}$), what are the possible values for $\det A$?

Hint: $\det(A^T) = \det(A^{-1}) \implies \det A = 1/\det A$.