00:00 Intro

01:05 Determinant in two Dimensions

02:10 Determinant and singular matrices

02:30 Proposition for non-singular matrices

04:41 Cramer’s Rule

07:30 Proof of Cramer’s Rule

12:27 Credits

Q1: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix with $\det(A) = 0$. Which claim is not correct?

A1: The columns of $A$ are linearly independent.

A2: The rows of $A$ are linearly dependent.

A3: $\mathrm{rank}(A) < n $

A4: $\mathrm{Ker}(A) \neq { 0 }$

A5: $A$ is not invertible.

A6: $A \mathbf{x} = \mathbf{b}$ has either no solution or infinitely many.

Q2: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix with $\det(A) \neq 0$. Which claim is correct?

A1: $A \mathbf{x} = \mathbf{b}$ has a unique solution for every right-hand side $\mathbf{b}$.

A2: The columns of $A$ are linearly dependent.

A3: $\mathrm{rank}(A) < n $

A4: $\mathrm{Ker}(A) \neq { 0 }$

A5: $A$ is not invertible.

Q3: Let $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \in \mathbb{R}^{2 \times 2}$ be a square matrix with $\det(A) \neq 0$. What is the second component of the solution of $A \mathbf{x} = \begin{pmatrix} b_1 \ b_2 \end{pmatrix}$?

A1: $x_2 = \frac{\det \begin{pmatrix} a_{11} & b_1 \ a_{21} & b_2 \end{pmatrix}}{\det \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}}$

A2: $x_2 = \frac{\det \begin{pmatrix} b_1 & a_{21} \ a_{21} & b_2 \end{pmatrix}}{\det \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}}$

A3: $x_2 = \frac{\det \begin{pmatrix} a_{11} & a_{21} \ a_{21} & b_2 \end{pmatrix}}{\det \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}}$

A4: $x_2 = \frac{\det \begin{pmatrix} a_{11} & a_{21} \ a_{21} & b_2 \end{pmatrix}}{\det \begin{pmatrix} b_1 & b_2 \ a_{21} & a_{22} \end{pmatrix}}$

A5: $A$ is not invertible.