Question

I need help understanding the process of how to get the answer to this matrices question please. The attachment is below.

Answer 1

@Directrix

Answer 2

if the matrix is singular it does not have an inverse.
For a matrix A to be singular, Det(A)=0.

Answer 3

Do you know how to calculate the determinant of your matrix?
If so, for what value of x will the determinant be 0?

Answer 4

Im not quite sure.

Answer 5

Do I set it up as a 3x2 matrix?

Answer 6

(you can only take the inverse of square-matrix (n by n matrix).)

Answer 7

I will give you an example problem, ok?

Answer 8

Ok

Answer 9

Oh whoops I was looking at the wrong question haha forget about the 3x2 matrix thing.

Answer 10

Question:

For what value of z does the matrix A have no inverse?
\(\color{#000000}{ \displaystyle A=\left[\begin{matrix}z & 6\\ 3 & 9\end{matrix}\right] }\)

\(\color{red}{\bf \text{--------------------------------------------------------}}\)

Solution:
If the determinant of a matrix is zero, then it is singular
and will have no inverse.

What is the determinant of matrix A?
\(\color{#000000}{ \displaystyle {\rm Det}(A)=z\cdot 9-6\cdot 3=9z-18 }\)

When is the determinant is zero (for inverse not to exist)?
\(\color{#000000}{ \displaystyle 9z-18 =0 }\)
\(\color{#000000}{ \displaystyle 9z=18 }\)
\(\color{#000000}{ \displaystyle z=2 }\)

So in this case, for x=2, the matrix A has no inverse.

Answer 11

I mean z=2, sorry.

Answer 12

okay Ill try and plug my numbers into that equation. One minute:)

Answer 13

Which equation ?

Answer 14

I mean not equation

Answer 15

But like Ill do that process

Answer 16

oh, alright ... go ahead, and take your time.
I will type an additional note on this.

Answer 17

is it -4?

Answer 18

yes, very nice

Answer 19

for x=-4 the inverse won't exist.

Answer 20

Oh okay I see now. Thank you very much!

Answer 21

Explanatory note, using the example:

In the previous example where
\(\color{#000000}{ \displaystyle A=\left[\begin{matrix}z & 6\\ 3 & 9\end{matrix}\right] }\)
you don't have an inverse for \(\color{#000000}{ \displaystyle z=2 }\).
But why?

Observe that \(\color{#000000}{ \displaystyle \left[\begin{matrix}2 \\ 3 \end{matrix}\right] }\) is a multiple of \(\color{#000000}{ \displaystyle \left[\begin{matrix}6\\ 9 \end{matrix}\right] }\),

that is: \(\color{#000000}{ \displaystyle 3\cdot \left[\begin{matrix}2\\ 3 \end{matrix}\right]=\left[\begin{matrix}6\\ 9 \end{matrix}\right] }\), so you can think

of it as if you have only one vector in there.

Answer 22

And you can't take an inverse of a vector in R^2.

Answer 23

Oh I get it now! Okay that makes a lot more sense haha thank you.

Answer 24

but, won't complicate here with column space and all that linear algebra.... we got the answer, and we know how to do the problem now...

won't ovewhelm.

Answer 25

good luck

Answer 26

Thank you very much:)