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(2) (10 points) Find(Extra credit, 5 points) Let 012 013 044 03 042 a2 021 02 024 B = a4 018 0J 031 032 03 024 Q4l 012 041 042 043 an Gp 021 be two antisymmetric ma...

Question

(2) (10 points) Find(Extra credit, 5 points) Let 012 013 044 03 042 a2 021 02 024 B = a4 018 0J 031 032 03 024 Q4l 012 041 042 043 an Gp 021 be two antisymmetric matrices, where 04k ~Oti, or AT = -A and BT B. Show tat AB = BA and present this diagonal matrix &s follows BA = AB = (a12414 + 013024 + 021C3) [, where [ iS the X 4-identity matrix: Find A-I and B-! . (H: Minkowski; 1908)

(2) (10 points) Find (Extra credit, 5 points) Let 012 013 044 03 042 a2 021 02 024 B = a4 018 0J 031 032 03 024 Q4l 012 041 042 043 an Gp 021 be two antisymmetric matrices, where 04k ~Oti, or AT = -A and BT B. Show tat AB = BA and present this diagonal matrix &s follows BA = AB = (a12414 + 013024 + 021C3) [, where [ iS the X 4-identity matrix: Find A-I and B-! . (H: Minkowski; 1908)



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$\operatorname{Let} B=\left[\begin{array}{rrr}11 & -8 & 4 \\ -8 & -1 & -2 \\ 4 & -2 & -4\end{array}\right]$ (a) Find all cigcnvalues of $B$ (b) Find a maximal set $S$ of nonzero orthogonal eigenvectors of $B$ (c) Find an orthogonal matrix $P$ such that $D=P^{-1} B P$ is diagonal.

So we need to show that Matrix A B C. D. Has There is the identity matrix. So we're going to go through the Matrix multiplication first, so we're going to multiply across and down. So zero point for a less 0.6 scene 0.3 a plus 0.2 c. They were gonna do the same thing with the second column. So 0.4 b plus 0.6 D 0.3 b plus 0.2 deep. Okay. And this will be equal to the Matrix that has 0.40 point 60.3 and 0.2. Okay, so we're gonna write a system of equations. The 0.4 a plus 0.6 scene is equal to 0.4 0.3 a plus 0.2 c is equal to 0.3. We're gonna right the other set of equations. Zero point for B plus 0.6 D 0.3 b plus 0.2 D is equal to 0.2 on the top on there is equal to 0.6. Okay, solving these equations we're going to need to will focus on the one that has A and C first. Okay, We need to figure out variable to eliminate. Let's choose to eliminate the sea first. So I'm gonna multiply the bottom equation by negative three. So the top equation stays the same. The bottom equation that when I multiply by negative three, I would end up with negative 30.9 A minus 0.6 c equals negative 0.9. And then I would add these two equations. Okay, so 20.4 minus 0.9 would be negative 0.5 A. The seas would cancel out in that would equal negative 0.5. So dividing by negative 0.5 A is equal to one. Now that I have a I need to find C so I plug a back in. So 0.4 times one plus 0.6 c equals 0.4. Uh, 0.4 times One is this 10.4 plus 0.6 c equals 0.4. Subtract the 0.4 to get 0.6 C is equal to zero, so C is equal to zero. Okay, we're gonna follow a similar set of steps on the other side here, so we need to choose to eliminate a variable. I'm gonna eliminate this, De. Just like I did on the last problem. Physical efficiency of stayed the same. It's just they have different letters. So we're gonna multiply the bottom equation by negative three. Okay, So negative 0.9 B minus 0.6 d equals negative 0.6. Adding these together we get negative 0.5 b, the D's would cancel out, and this would equal zero, so B is equal to zero. Now that I have B, I need d. So 0.4 times zero plus 0.6 d equals 0.6. So, zero that would go away. So I'd have 0.6 d equals 0.6. So do you Zeke would have won if we plug these numbers back into the matrix. That was given that we were trying to solve for Major. Her position A was one be with zero CIA zero India's. Well, okay, so I end up with the identity matrix

This problem we've been given matrix A. Is equal A. B, C. D. And matrix B. There's a Matrix 1001. And we would like to find the value of A. B. And value of B. A. Find A B. This is A. B C. D. Times 1001. We multiply the rose by the columns. We have a plus zero. And that's a. Now if we move over and multiply it by the second column, this will be zero. B is B. Now let's move down. And multiply second role by the first column. B, C plus zero is C. And then the last problem we have zero plus D. Is D. And so this is the matrix A times B. Now that we also want the other way around. So we won't be times A. Just 1001 times A. B C. D. And again we're gonna multiply the row by the calm. And so we have a policy zero which is A. And then we move over. Be paused zero, which is B. Move down zero plus. You would just see we have already have zero plus D. Is deep. My answer here should be A. B. C.


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