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Verify Caley-Hamilton Theorem for the matrix A =hence find 4-1...

Question

Verify Caley-Hamilton Theorem for the matrix A =hence find 4-1

Verify Caley-Hamilton Theorem for the matrix A = hence find 4-1



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Verify Theorem 1.7 .4 for the given matrix $A .$ (a) $A=\left[\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}\right]$ (b) $A=\left[\begin{array}{rrr}1 & -2 & 3 \\ -2 & 1 & -7 \\ 3 & -7 & 4\end{array}\right]$

Were asked to find the null space of egg. The definition of the north face is shown up top. We have that. The north face of an n by N Matrix A is equal to the solution set with the vector X with elements in or M which satisfies the homogenous linear system. A X equals zero and the problem gives a has a one by two matrix 14 So we can see from that that our in is equal to So we're in our to space, which means are expect er will be the two by one matrix which will look like this x one x two. Now we'll write out our X equals zero so we have or matrix a vector X, which is equal to one for times X one, the next two. Are you expecting what's blowing that out? We have one times x one plus four times X two is equal to zero now solving for X one we can get X one is equal to negative for times next to now we can just let this x two equal any other variable. We will say t so let x two. Nick will see that's now we can rewrite this as X one is equal to minus four times team. Now we can right this out in a solution. Set. Do you recall what that looks like? A top here, The north space. So we'll have the the North place. Well, that is equal to the solution. Set our vector X and so that's our two by one matrix when we have that are ex warm zero K minus 14 in our X two is equal T and that's such that t is made of ah, riel elements. So this is the answer that is the north space of a

Mhm. All right. We would like to identify if A is convertible. So to do that, they tell us to find the determinant and the determine of A two by two matrix is written right over here, so are determinant of A is going to equal four times forward to 16 minus five times three, which is 15. Are determinant is one. So what does that tell us? Well, it's not equal to zero. That means A. Is convertible. So now I can find the inverse of A.

So in the following problem, we want to show that a is the embers matrix of me. So in other words, we want to show that a physical to be members. One of the properties of this matrices, if that is a is the university. Then a times B must equal I subdue, which is the identity matrix off actuator metrics, which I said too. It's just equal to 1100 So then we were gonna prove that. Mr Multiply a times B, we have a matrix 457 Sirrah multiplied by 01 over seven won over five and negative for the vote of a 35. So we're gonna move to play this first row by this first column, so we'll have four times zero and then five times one of her five. Next, we're gonna multiply this first throw by the second column so we'll have four attempts one over seven plus five times native for regrettably 35. Now we're gonna multiplied this second row by the first columns will have seven time zero plus syrup terms. One of her five. Then we're going to multiply the second row by the second column so we'll have seven times one over seven plus zero attempts to negative four provided by 35. This time right here is zero is when it's also zero when he stole the serum and this is also zero. So we're gonna simplify this. So five divided by one of her five is just one four times. One over seven is just four divided by seven. And if we take a five over here, we have that. This is just equal. Your negative four divided by seven. Then this term is just your own. And then seven times one over seven it's just one. And so distant right here is also zero. So we have shown that eight times be physical to the identity matrix. Refrigerate your column. Which means that a a sequel to be embers and that be physical to a embers. And this is the solution to the question

And this question you are given a matrix A Which is equal to a B. See minus. We have to find a situation in which the matrix is in military for that purpose will make to find a square which is a multiple. Our air. So we'll make it by me to say with itself A B c minus A symmetric say multiplied by matrix which is a B C minus air. Now we will multiply rosa first metrics with columns of second metrics with our A. We get a square Damon. Let me see. You get a B. C. Mm. Primary we get a B merely by minus A. We get minus a B. No seam with a We get to see A. My son like I see you get C. A. Sounds like maybe we get C. B. When I say when I say we get this year square. So when we saw this, you get this grade is good too. Escape this Bc zero. CNC considered zero and escape this. B C. Yeah. For this matrix to be in the military, Yes, we must be equal to identity matrix. So a square is a scripless Bc 00 Sk Plus BC. This must be There too. Identity matrix of same order, which is 1001. Just from the definition of government. This is corresponding with mystical, so a squared less. Listen the speaker to mine. So this is the condition which makes metrics the military, which is the answer of this question. Thank you for watching me. Do


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