# Lecture - Linear Algebra MT22, III

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/3/ · Updated: 2022-10-25 · Tags: uni, lecture

### Flashcards
What does it mean if Gauss elimination on $A \in \mathbb{R}^{n\times n}$ gives an upper trianglar matrix with a non-zero diagonal?::
$A^{-1}$ exists.

What does it mean if $Ax = 0$ only has one solution, $x = 0$ (for matrices)?::
$A$ is invertible.

What does it mean if Gauss-Jordan elimination on $[A | I]$ yeilds $[I | B]$?::
$A^{-1} = B$

What would be the $3 \times 3$ matrix for the ERO corresponding to swapping rows 2 and 3?::
$$
\left(\begin{matrix} 1 \& 0 \& 0 \\\\ 0 \& 0 \& 1 \\\\ 0 \& 1 \& 0 \end{matrix}\right)
$$

What would be the $3 \times 3$ matrix for the ERO corresponding to multiplying row 3 by a constant $\lambda$?::
$$
\left(\begin{matrix} 1 \& 0 \& 0 \\\\ 0 \& 1 \& 0 \\\\ 0 \& 0 \& \lambda \end{matrix}\right)
$$

What would be the $3 \times 3$ matrix for the ERO corresponding to adding $\lambda$ lots of row 1 onto row 2?::
$$
 \left(\begin{matrix} 1 \& 0 \& 0 \\\\ \lambda \& 1 \& 0 \\\\ 0 \& 0 \& 1 \end{matrix}\right) 
$$

What is true about the product of two invertible matrices $AB$?::
It is also invertible.

What happens to the element $a_{ij}$ of a matrix when it is transposed?::
It becomes $a_{ji}$.

What is $(A + B)^\top$?::
$$
A^\top + B^\top
$$

What is $(A^\top)^{-1}$?::
$$
(A^{-1})^\top
$$

What does it mean for a matrix to be symmetric?::
$$
A = A^\top
$$

What does it mean for a matrix to be skew-symmetric?::
$$
A^\top = -A
$$

What is true about the diagonal entries of a skew-symmetric matrix?::
They are zero.

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