Algebra

Example of finding matrix inverse | Matrix transformations | Linear Algebra | Khan Academy

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We stumbled upon the sword video in determining the inverse of an inversible matrix Consequently, we will actually use the inverse finding method in this video Also, I’m going to use the same matrix that we started with in the previous video which looks kind of like a good matrix We know that the reduced row formula form of a matrix is \u200b\u200ba unit matrix and therefore we know that this matrix is \u200b\u200breversible. And for this, let’s find the inverse The mechanism of finding excises is fairly easy. It is just doing the same transforms that you use for this matrix to get the unit matrix. You also apply these same transforms to the unit array This is because all of these transforms, if represented as matrices, are the inverse of this matrix. Let’s do that. I’m going to create an extra matrix here, I’ll try to make it more organized here First, I will write a which is (one, negative one, one) then (one, two, one) (negative one, three, four) Then I will extend this matrix to the unit matrix with (one, zero, zero, zero, one, zero, zero, zero, one) Now, if we wanted to get the reduced row score form, we would replace the third row I will keep the first row as it is now, I will draw it here Total first row: One, one negative, one negative. Will be provided with (one, zero, zero) And we will keep the first row as it is. However, we will replace the second row in the second row plus the first row So we have a minus one plus one equals zero And two plus minus one is equal to one And three plus minus one one equals two Zero plus one equals one And one plus zero equals one And finally zero plus zero equals travel And all I did now was I combined these two rows. We want to get a class here. Thus I will replace the third row with the third row minus the first row So, one minus one equals zero And one minus minus one equals two And four minus one minus is five Zero minus one equals negative one And zero minus zero equals zero Finally, one minus zero equals one. And so on Now what do we want to do? Well, we expanded a lot We want to equal both this and that element by zero So, we will keep the second class the same. I will write it here (zero, one, two) and then we supply it with one, one, zero) And I’ll replace the first row with the first row plus the second row So we have one plus zero equal one And minus one plus one equals zero Since I did that, to get zero here Minus one plus two equals one And one plus one equals two And zero plus one limit equals And zero plus zero equals zero Now, I want to yellow this element here Thus, we will replace the third row with the third row minus two times in the second row And so we have, zero minus two times zero is zero And two minus two multiplied by one equals zero Five minus two multiplied by two equals five minus five minus four equals one Minus one minus two multiplied by one … then minus one minus two equals three Zero minus two times one is equal to minus two Then, one minus two multiplied by zero equals one Well, we have finished the last stage And now I want to yellow these items here. So, I will keep the third row as it is I will change colors in order to diversify the colors We’ll have this row (zero, zero, one) We will also supply it with minus three, minus two and one) Now, I will replace the first row with the first row minus the third row So we’ll have: one minus zero is equal to one Zero minus zero equals zero And one minus one equals zero And two minus minus three equals five And one minus minus two equals three Finally, zero minus one equals one Now, we will replace the second row with the second row minus two times the third row So we’re going to have zero minus two times zero equals zero And one minus two multiplied by zero equals zero And two minus two multiplied by one equals We should be careful not to make a mistake here Zero minus two multiplied by zero equals zero And one minus two multiplied by zero equals one Two minus two multiplied by one equals zero And one minus two multiplied by negative three … and this one plus two multiplied by three … equals seven One minus two is multiplied by minus two and this equals one plus four equals five Then, zero minus two is multiplied by one and this equals two Likewise, we have made Part A of our increased matrix in the reduced row grade model Thus this is a reduced grade grade model for Part A Thus, when applying these same transfers Thus, when applying these same transforms to the unit matrix, you get the inverse of the matrix A And this is the inverse of matrix A Now, we created the inverse easily

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