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1. Lct A, B and € bc matriccs, with det(4) = 7, dct(B) = and det(C) =-5. Find cach of thc following quantitics and answcr thc accompanying qucstions _ Your wo...

Question

1. Lct A, B and € bc matriccs, with det(4) = 7, dct(B) = and det(C) =-5. Find cach of thc following quantitics and answcr thc accompanying qucstions _ Your work should clcarly justify thc rcsult using thc dctcrminant_ propertics rclatcd t0 matn- multiplication:det(30). must bc clcar how this can bc detcrined from thc givcn valuc of det(C) matrix multiplication and rclatcd dctcrminant propcrtics .det(A-!) det(B -= and det(4-'C-I)If CX = A, what is det( X)?

1. Lct A, B and € bc matriccs, with det(4) = 7, dct(B) = and det(C) =-5. Find cach of thc following quantitics and answcr thc accompanying qucstions _ Your work should clcarly justify thc rcsult using thc dctcrminant_ propertics rclatcd t0 matn- multiplication: det(30). must bc clcar how this can bc detcrined from thc givcn valuc of det(C) matrix multiplication and rclatcd dctcrminant propcrtics . det(A-!) det(B -= and det(4-'C-I) If CX = A, what is det( X)?



Answers

Find (a) $\operatorname{det}(A),$ (b) the matrix of cofactors $M_{C},(\mathrm{c})$ adj $(A),$ and, if possible, $(\mathrm{d}) A^{-1}.$ $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & -1 & 4 \\ 5 & 1 & 7 \end{array}\right]$$

Hello. So here we can take our two matrices A. B C. D. And D. Negative be negative. See negative D. Negative be negative. See a multiply those together to give us while we get an A D minus B. C. And then our entry here becomes a negative A. B plus a beach. That's zero. And the entry here becomes a C. D minus C. D. So that's zero. And then we get a negative B C plus A. D. Okay and then um if we multiply that by one over a D minus B C. We get one over a um a d minus Bc times the matrix here A D minus B C. 00 And a negative B. C plus A. D. Um is going to give us um just the identity. So this is going to give us +1001 So therefore one over a D minus Bc times a B C. D. Times the negative be negative C. A. Is going to be equal to the two I two identity here +10 P 01

This is a question number 46 1st of all, find the determinate off the metrics. Severe disease will do You dance here? Negative too. Minus seven less. Three times your one minus one multiple. Every minus one. There is minus one and zero plus five times. Your sound murdered by one that is seven minus zero of your days equals two times negative. Nine. Last three time. Negative one and fight. I'm seven. So Dini's negative rating negative. Three plus 35 right? Therefore, the German Andy's or Dean right now find all minors off the medics. So here, Memorable one, Which means eliminate first row and for school. So your enviable one is 21 still in minus one in which is recorded Negative nine. Now I m one do that is eliminated. First row and second column. So here, one one year zero. And here. Negative one, right? No, I m one through that is eliminated first role and told column that he's one, too. And 07 which is equal to seven. Now. I am not one more. That is 21 which means I live in a second row and first column right with your endure knees. Negative three by so in minus one, which is equal to negative. 22. No, I am. Double do that is eliminated. Second row and second column. Right, you hear? 25 and zero minus one is equal to minus two. Now I am tutoring right, which means eliminated second row and column. So m two trees to minus 307 which is equal to 14. No entry. One that is Eliminate taller Drawing first column of your negative five. Here to one video recorder. Negating. Starting. No empty. Remembering studies 25 and 11 Logistical doing 83. Now I am number three. That is two minus three. Want to video record with seven? Now write all miners in the medics. Right. So here are many do nine negative. One say one negative. 20 to negative. Two. 14. You're negative. According negative three and seven. Then you've never met. Takes off minus right now, Jane. The sign of the mid range According to sign matrix. So begins. Have that we have toe only change the sign off. Well minus one. Minus 32. Your negative three and 14. Right. By changing this four elements, sign that gives the cough vector matrix and see Right. This is our perfecter, my bricks EMC now a joint off issue. And by transpose old m c. Right, So take France Borgoff m c Your negative 917 They told you to minus two negative affording and your negative holding three and seven. This is a joint off a now in north metrics is doing by one by d multiply by and joint. Okay, Now our determinant off metrics is 14 on the joint off A that he's here. Negative 917 here to do minus two. And here for dean your negative. Toting three and seven. Right now the investment makes you're mainly due nine divided by 14 thirties No animated by 40 Corden Directed by for being dirty 16 devoted base seven Your negative told in doing it by 40 one by fording Nobody won by 73 by 14 one by two minus one and one by two So this is over a in worse Thank you

This is a question number 40 1st of all, find e determinant of the medics. So here, minus one, Marty, by by one that is minus one and your four multiplied by minus two, that is minus eight. So you're easy. Quarter minus one. Therefore, the communities seven, right, No find. And the bottle on which means eliminate first royal for school. So your end about one is one, similarly and want to his four and 21 is negative. Two on and about Louise. Negative one. Right now. Write all these foreign tears in the Matrix. You're one foreign negative to 91. Right? That gives the medics off minors right now. Jane, the sign off the med makes according toa last minus minus plus. Right So far. Medics off protector, that is N C. Ease. Year one minus 40 or negative? Negative. Was it used here too? The year 1981. But these this is the conflict or metrics and see right, well, joint off a issue. And by transpose off and see right, But he all right, trying off a unit for do here, you know emcees one minus four. So our colonies negative one minus four Now The second Roy's Tu minus one, right through our second Balinese to minus one. Right. But this is our joint off. A no endorsement makes is you. And by one, by determinant, off a multiplied by I joined all fate like So you're the easy called 07 Find the majorities. Want toe? No go do for Mega Do you want so in worse matrix these one place they wouldn't. You're grew by seven your integrity for by seven and your negative one right there. This is our investment weeks. Thank you.

And this problem were given a matrix a B C D, which is equal to three identity matrix 1001 And we're multiplying that by, um, this matrix right here. Nine negative seven negative five and four. And so because this has two rows and two columns, our average our final matrix is going to have she rose in two columns. And so A is going to be equal to a matrix with four entries, two on the top and two on the bottom. And so I'm going to begin by writing out R A B c d. Because we know that that is equal to 1001 And so to multiply the matrices, we're going to start with this first top left country right here and is going to be equal to nine times one plus negative seven times zero. We got that by taking the first row in the first column of each respective matrix. And so then we're gonna do the same thing for the next entry. We're gonna take nine times zero this time Organized the second column 9 10 0 plus negative seven times one and then we're going to repeat this process for the next two entries, The sun is going to be the bottom row on the first column. So negative five times one plus four times zero. And then lastly, we have the bottom row in the second column Negative five times zero plus four times one. And so we can see that the zeros all cancelled out and they're all equal to zero. Which means we're only left with the entries. I have, ah, one better multiplied by one. So nine negative seven negative five and four which you can see is the same thing that we multiplied the identity matrix by. So this property shows us that if you have a matrix multiplied by the identity matrix, it's going to be equal to that original matrix.


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