even more linear algebra notes

there only exists 1 inverse matrix for every invertible matrix

$\text{[math]}$

transposition properties

$\text{[math]}$

the proof is possible using induction

if matrix $\text{[math]}$ is square and its possible to arrive at a matrix with a row of 0's using elementary operations on $\text{[math]}$ then $\text{[math]}$ isnt invertible

if $\text{[math]}$ there is a solution for every $\text{[math]}$ then $\text{[math]}$ is invertible

if $\text{[math]}$ is square and its possible to arrive at $\text{[math]}$ using elementary operations then it has a single solution

$\text{[math]}$ example:
$\text{[math]}$

prove $\text{[math]}$

prove if $\text{[math]}$ and $\text{[math]}$ then for every $\text{[math]}$ its true that $\text{[math]}$

proof using induction

if for $\text{[math]}$ we get $\text{[math]}$ then for every $\text{[math]}$ we get $\text{[math]}$

proof using induction

prove $\text{[math]}$ , assuming the multiplication is possible

$\text{[math]}$

prove that matrix multiplication is distributive, $\text{[math]}$ , $\text{[math]}$

$\text{[math]}$ proof:
$\text{[math]}$

given that $\text{[math]}$ is invertible prove that $\text{[math]}$ if and only if $\text{[math]}$

if $\text{[math]}$ then $\text{[math]}$

$\text{[math]}$

if $\text{[math]}$ then $\text{[math]}$

$\text{[math]}$

given $\text{[math]}$ is a square matrix, prove $\text{[math]}$ is symmetric

$\text{[math]}$

given $\text{[math]}$ , prove $\text{[math]}$ and $\text{[math]}$ are invertible and $\text{[math]}$

$\text{[math]}$ so $\text{[math]}$ and $\text{[math]}$ are inverse of each other
$\text{[math]}$ so $\text{[math]}$ and $\text{[math]}$ are inverse of each other
and so $\text{[math]}$ and $\text{[math]}$ are both invertible
$\text{[math]}$

is $\text{[math]}$ correct?

proving $\text{[math]}$ and $\text{[math]}$ are invertible:
$\text{[math]}$ proving the given formula:
$\text{[math]}$

given $\text{[math]}$ , prove $\text{[math]}$ is invertible and find its inverse

$\text{[math]}$

prove that if $\text{[math]}$ is invertible then $\text{[math]}$

$\text{[math]}$

prove $\text{[math]}$

$\text{[math]}$

prove $\text{[math]}$

$\text{[math]}$

prove that if $\text{[math]}$ is singular then $\text{[math]}$

$\text{[math]}$

given $\text{[math]}$ and $\text{[math]}$ are invertible, prove $\text{[math]}$

$\text{[math]}$

prove that for every 2 matrices $\text{[math]}$ the following is true: $\text{[math]}$

whatever

prove that if $\text{[math]}$ and $\text{[math]}$ are invertible matrices with the size $\text{[math]}$ then $\text{[math]}$ is also invertible and $\text{[math]}$

$\text{[math]}$ and so $\text{[math]}$ is the inverse of $\text{[math]}$

prove that $\text{[math]}$ is invertible if and only if $\text{[math]}$ has a single solution

this equation only has 1 solution when $\text{[math]}$ because if $\text{[math]}$ had a
row with zeros we would get infinite solutions
and since $\text{[math]}$ doesnt contain a row of zeros then it must be invertible

given the matrices $\text{[math]}$ with the dimensions $\text{[math]}$ with the following

$\text{[math]}$ prove $\text{[math]}$ are invertible and find $\text{[math]}$ for $\text{[math]}$
we can join the 2 given equations into 1
$\text{[math]}$ finding $\text{[math]}$ (need to prove $\text{[math]}$ is invertible):
$\text{[math]}$

prove if $\text{[math]}$ is invertible then $\text{[math]}$ are both invertible

proof $\text{[math]}$ is invertible
$\text{[math]}$ proof $\text{[math]}$ is invertible
$\text{[math]}$

$\text{[math]}$ is a column vector in $\text{[math]}$ and $\text{[math]}$

prove $\text{[math]}$ is a symmetric matrix

prove if $\text{[math]}$ then $\text{[math]}$ is self-invertible

prove that if we can get from $\text{[math]}$ to $\text{[math]}$ using a finite set of elementary operations then $\text{[math]}$ is invertible

$\text{[math]}$

complex plane formulas

$\text{[math]}$

if $\text{[math]}$ is singular then $\text{[math]}$

singular = no inverse
$\text{[math]}$

prove $\text{[math]}$

for the multiplication to be possible the matrices have to have the
following dimensions
$\text{[math]}$ proof:
$\text{[math]}$

there only exists 1 inverse matrix for every invertible matrix

transposition properties

if matrix is square and its possible to arrive at a matrix with a row of 0's using elementary operations on then isnt invertible

if there is a solution for every then is invertible

if is square and its possible to arrive at using elementary operations then it has a single solution

prove

prove if and then for every its true that

if for we get then for every we get

prove , assuming the multiplication is possible

prove that matrix multiplication is distributive, ,

given that is invertible prove that if and only if

if then

if then

given is a square matrix, prove is symmetric

given , prove and are invertible and

is correct?

given , prove is invertible and find its inverse

prove that if is invertible then

prove

prove

prove that if is singular then

given and are invertible, prove

prove that for every 2 matrices the following is true:

prove that if and are invertible matrices with the size then is also invertible and

prove that is invertible if and only if has a single solution

given the matrices with the dimensions with the following

prove if is invertible then are both invertible

is a column vector in and

prove is a symmetric matrix

prove if then is self-invertible

prove that if we can get from to using a finite set of elementary operations then is invertible

complex plane formulas

if is singular then

prove

if matrix $\text{[math]}$ is square and its possible to arrive at a matrix with a row of 0's using elementary operations on $\text{[math]}$ then $\text{[math]}$ isnt invertible

if $\text{[math]}$ there is a solution for every $\text{[math]}$ then $\text{[math]}$ is invertible

if $\text{[math]}$ is square and its possible to arrive at $\text{[math]}$ using elementary operations then it has a single solution

prove $\text{[math]}$

prove if $\text{[math]}$ and $\text{[math]}$ then for every $\text{[math]}$ its true that $\text{[math]}$

if for $\text{[math]}$ we get $\text{[math]}$ then for every $\text{[math]}$ we get $\text{[math]}$

prove $\text{[math]}$ , assuming the multiplication is possible

prove that matrix multiplication is distributive, $\text{[math]}$ , $\text{[math]}$

given that $\text{[math]}$ is invertible prove that $\text{[math]}$ if and only if $\text{[math]}$

if $\text{[math]}$ then $\text{[math]}$

if $\text{[math]}$ then $\text{[math]}$

given $\text{[math]}$ is a square matrix, prove $\text{[math]}$ is symmetric

given $\text{[math]}$ , prove $\text{[math]}$ and $\text{[math]}$ are invertible and $\text{[math]}$

is $\text{[math]}$ correct?

given $\text{[math]}$ , prove $\text{[math]}$ is invertible and find its inverse

prove that if $\text{[math]}$ is invertible then $\text{[math]}$

prove $\text{[math]}$

prove $\text{[math]}$

prove that if $\text{[math]}$ is singular then $\text{[math]}$

given $\text{[math]}$ and $\text{[math]}$ are invertible, prove $\text{[math]}$

prove that for every 2 matrices $\text{[math]}$ the following is true: $\text{[math]}$

prove that if $\text{[math]}$ and $\text{[math]}$ are invertible matrices with the size $\text{[math]}$ then $\text{[math]}$ is also invertible and $\text{[math]}$

prove that $\text{[math]}$ is invertible if and only if $\text{[math]}$ has a single solution

given the matrices $\text{[math]}$ with the dimensions $\text{[math]}$ with the following

prove if $\text{[math]}$ is invertible then $\text{[math]}$ are both invertible

$\text{[math]}$ is a column vector in $\text{[math]}$ and $\text{[math]}$

prove $\text{[math]}$ is a symmetric matrix

prove if $\text{[math]}$ then $\text{[math]}$ is self-invertible

prove that if we can get from $\text{[math]}$ to $\text{[math]}$ using a finite set of elementary operations then $\text{[math]}$ is invertible

if $\text{[math]}$ is singular then $\text{[math]}$

prove $\text{[math]}$