How would I solve the following?

sktsasus

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Sep 17, 2017
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41
1 -4
-6 -6


Note: The set of numbers above is the matrix M.


Find c1 and c2 such that M^2 + c1M + c2I2 = 0, where I2 is the identity 2 x 2 matrix.

I got stuck after I merged c1 and c2 into their respective matrices and calculated the square of M.

By rearranging the equation (after merging c1 and c2 and squaring M), I got:

c1 -4c1 + c2 0 = -1 -16 This gave me a c1 value of 4 and c2 value of -12. Neither of which were correct.
-6c1 -6c1 0 c2 -36 -36
 
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The set of steps you've outlined seems correct to me, so any errors must have occurred in the matrix multiplication and/or addition. It looks to me like you found that M2 is this matrix:

\(\displaystyle \begin{pmatrix}1&-4\\ \:-6&-6\end{pmatrix}^2=\begin{pmatrix}1&16\\ 36&36\end{pmatrix}\)

This result suggests you've interpreted squaring a matrix to just mean squaring each individual term, which is absolutely not correct because that's not what squaring means with respect to matrices. Let's briefly return to the definition of squaring to see why this is wrong. When working with ordinary real numbers, we know that x2 = (x)(x). In words, squaring x means to multiply x by itself. And the exact same definition applies to matrices, so what happens if we apply that?

\(\displaystyle \begin{pmatrix}1&-4\\ \:-6&-6\end{pmatrix}^2 = \begin{pmatrix}1&-4\\ \:-6&-6\end{pmatrix} \: \begin{pmatrix}1&-4\\ \:-6&-6\end{pmatrix}=\begin{pmatrix}?&?\\ ?&?\end{pmatrix}\)
 
What does "merging c1 and c2" mean?

Can you solve x^2 + bx + c = 0 for x?
 
@ksdhart2 Yes, you are right. My squaring of the matrix M was completely incorrect. I have solved it now. Thank you.

@tkhunny Thank you as well.
 
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