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242LN1.5 - 1.5 Inverses of Matrices There are actually two...

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1.5. Inverses of Matrices* There are actually two things we need to do, to make the X = B A formula usable. Find a matrix I such that IM = M for all matrices M (where the product is defined); Given a matrix A , find a matrix C so that CA = I. Then our idea will go through . . . * And one last part of Section 1.4.

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AX = B C ( AX ) = CB ( CA ) X = CB IX = CB X = CB
So which matrix will make IM = M for all matrices M ? (In particular, what if M = 1 - 1 2 2 - 2 8 1 - 2 0 ?)

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So which matrix will make IM = M for all matrices M ? (In particular, what if M = 1 - 1 2 2 - 2 8 1 - 2 0 ?) What are the dimensions of I ? I · M = M ? × ? 3 × 3 3 × 3
So which matrix will make IM = M for all matrices M ? (In particular, what if M = 1 - 1 2 2 - 2 8 1 - 2 0 ?) What are the dimensions of I ? I · M = M ? × = × 3 3 × 3

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So which matrix will make IM = M for all matrices M ? (In particular, what if M = 1 - 1 2 2 - 2 8 1 - 2 0 ?) What are the dimensions of I ? I · M = M × 3 3 × 3 × 3 So I must be a 3 × 3 matrix. How should we fill in the entries?
So which matrix will make IM = M for all matrices M ? (In particular, what if M = 1 - 1 2 2 - 2 8 1 - 2 0 ?) No, 1 1 1 1 1 1 1 1 1 doesn’t work . . . 1 1 1 1 1 1 1 1 1 · 1 - 1 2 2 - 2 8 1 - 2 0 = 4 - 5 10 4 - 5 10 4 - 5 10

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So which matrix will make IM = M for all matrices M ? (In particular, what if M = 1 - 1 2 2 - 2 8 1 - 2 0 ?) However, 1 0 0 0 1 0 0 0 1 · 1 - 1 2 2 - 2 8 1 - 2 0 = 1 - 1 2 2 - 2 8 1 - 2 0 (In fact, this is the only such matrix that works.)
In general, the m × m matrix I m = 1 0 · · · 0 0 0 1 · · · 0 0 .

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242LN1.5 - 1.5 Inverses of Matrices There are actually two...

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