matrices

[jsbeckton]

Im stuck on a question concering matricies.

Let A be a 2x2 matrix such that A^2 = 2A . Find all possible values for det (A)

Will anything else besides the 0(2x2) and 1(2x2) matricies satisfy this? Leaving 0 as the only possible determinate?

 

[pka]

Any 2x2 matrix A such that det(A)=4 will satisfy the condition:
\(\displaystyle A = \left[ {\begin{array}
- 1 & - 1 \\
5 & 1 \\
\end{array}} \right]\)

 

[jsbeckton]

but A^2 must equal 2A, that matrix does not satisfy that requirement.

 

[pka]

You are correct.
I read it as det(A<SUP>2</SUP>)=det(2A).
Sorry.

P.S.
\(\displaystyle A^2 = 2A \Rightarrow \;A(A - 2I) = 0 \Rightarrow \;A = 2I{\rm{ or }}A = 0\)

 

[jsbeckton]

thanks anyways, I haven't figured out how to properly place exponents.

 

[stapel]

Letting a, b, c, and d be the entries of A, multiplying out A² and 2A, and equating the results, I get the following cases:

. . . . .b = 0, in which case a = 2.

. . . . .b <> 0, in which case a + d = 2.

. . . . .c = 0, in which case d = 2.

. . . . .c <> 0, in which case a + d = 2.

That is, if b = 0, then a = 2. If c = 0 also, then d = 2.

If b = 0, then a = 2. If c <> 0, then, since a + d = 2 and a = 2, then d = 0.

If c = 0, then d = 2. If b = 0 also, then a = 2.

If c = 0, then d = 2. If b <> 0, then, since a + d = 2 and d = 2, then a = 0.

This gives you four matrices, fully determined. Now the messy part....

If b <> 0 and c <> 0, then all we know is that a + d = 2. But if you try just picking anything for a and d, as long as a + d = 2, you'll find that things don't work out right. So there must be something else to it.

Letting "2 - a" stand in for "d", I get:

. . . . .a² + bc = 2a

Since we know bc <> 0 (since neither b nor c is zero), then:

. . . . .a² - 2a + bc = 0

. . . . .a = [2 ± sqrt(4 - 4bc)] / 2

. . . . . . .= [2 ± 2sqrt(1 - bc)] / 2

. . . . . . .= 1 ± sqrt(1 - bc)

I don't know how helpful this is.... :shock:

Eliz.

 

[jsbeckton]

anything is helpful, however I don't know what <> stands for?
eg: "If c <> 0".....

 

[stapel]

The opposite of "=". :wink:

Eliz.

 

[jsbeckton]

thanks, im used to "!="

 

[stapel]

You'll sometimes also see "=/=", a simulation of the "slash-equals" symbol.

Eliz.

 

[jsbeckton]

A^2 = 2A => A(A-2I)=0 => A=2I or A=0

seems logical but dosent include 1(2x2) which does satisfy the requirement.