Matrices with unknown elements

Question

Now trying to solve matrices.

If

$$A = \left[\begin{array}{rrr} -1&b\\ c&-1\\\end{array}\right]$$

What are the elements b and c so that $$A^{2}=2A^{T}$$

How to solve this kind of exercise? What are the step by step instructions for this kind of exercise?

Compute $A^2$ and $2A^T$ and compare the elements of both matrices. They should be equal to one another. — Math Lover, Sep 18 '17 at 21:25

score 1 · Answer 1 · answered Sep 18 '17 at 21:25

1

Hint

$1)$ $A^2=A\cdot A$

$2)$ $A^T = \left[\begin{array}{rrr} -1&c\\ b&-1\\\end{array}\right]$

$3)$ $2A^T = \left[\begin{array}{rrr} -2&2c\\ 2b&-2\\\end{array}\right]$

answered Sep 18 '17 at 21:25

Arnaldo

I got
$$A^{2} = \left[\begin{array}{rrr} bc+1&-2b\ -2c&bc+1\\end{array}\right]$$

and

$$2A^{T} = \left[\begin{array}{rrr} -2&2c\ 2b& -2\\end{array}\right]$$

But what now, how to continue? How should I interpret this?
– MiMaKo Sep 18 '17 at 21:50
@MiMaKo: Now each term must to be equal. For example, $bc+1=-2$, $-2b=2c$, ... – Arnaldo Sep 19 '17 at 12:27
Thanks a lot. But how to solve b and c from this equation? (Sorry, It's been many years I studied maths, so I don't remember anymore). I get:
$$bc + 1 = -2$$

$$bc = -3$$

So now I know that bc is -3, but how to get the solution for only b and only for c?
– MiMaKo Sep 20 '17 at 12:28
@MiMaKo: You also know that $-2b=2c\to b=-c$ and then, using your last expression, $c^2=3\to c=\pm \sqrt{3}$. – Arnaldo Sep 20 '17 at 13:29
Oh! Thanks a lot. I actually got that same solution but felt so unconfident with this exercise. So thank you for the support! – MiMaKo Sep 20 '17 at 16:14
@MiMaKo: you are very welcome! – Arnaldo Sep 20 '17 at 16:41

score 0 · Accepted Answer · answered Sep 18 '17 at 23:22

After you solve for both A² and 2A^T, you can set each index equal to each other and solve for b and c that way.

-2 = bc + 1
2c = -2b

And so on with all the indexes. Since each term will contain b and c, use substitution to solve for one then the other.

2 Answers2