Lecture - Linear Algebra MT22, III

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 $[A \vert I]$ yeilds $[I \vert B]$?

$A^{-1} = B$

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.