Theorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.

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1 Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse matrix explaatio: By the previous theorem, whe A exists, the solutios to liear systems A x b are uique (x A b From our discussios o Tuesday ad Wedesday, we kow that for square matrices, solutios to such liear systems exist ad are uique oly if the reduced row echelo form of A is the idetity (Do you remember why?) Thus by logic, wheever A exists, A reduces to the idetity I this case that A does reduce to I, we search for A as the solutio matrix X to the matrix equatio A X I ie A col X col X col X Because A reduces to the idetity matrix, we may solve for X colum by colum as i the examples we just worked, by usig a chai of elemetary row operatios: A I I B, ad deduce that the colums of X are exactly the colums of B, ie X B Thus we kow that A B I To realize that B A I as well, we would try to solve B Y I for Y, ad hope Y A But we ca actually verify this fact by reorderig the colums of I B to read B I ad the reversig each of the elemetary row operatios i the first computatio, ie create the chai B I I A so B A I also holds (This is oe of those rare times whe matrix multiplicatio actually is commuative) To summarize: If A exists, the solutios x to A x b always exist ad are uique, so the reduced row echelo form of A is the idetity If the reduced row echelo form of A is the idetity, the A exists That's exactly what the Theorem claims

2 There's a ice formula for the iverses of text sectio 36 o determiats: Theorem: this case, exists if ad oly if the determiat D ad matrices, ad it turs out this formula will lead to the ext bc of is o-zero Ad i d b ad bc c a (Notice that the diagoal etries have bee swapped, ad mius sigs have bee placed i frot of the offdiagoal terms This formula should be memorized) Exercise 8a) Check that this formula for the iverse works, for D (We could have derived it with elemetary row operatios, but it's easy to check sice we've bee haded the formula) 8b) Eve with systems of two equatios i two ukows, uless they come from very special problems the algebra is likely to be messier tha you might expect (without the formula above) Use the magic formula to solve the system 3 x 7 y 5 5 x 4 y 8 Remark: For a matrix, the reduced row echelo form will be the idetity if ad oly if the two rows are ot multiples of each other If a, b are both o-zero this is equivalet to sayig that the ratio of the first etries i the rows, the ratio of the secod etries Cross multiplyig we see this is the same as ad bc, ie ad bc This is also the correct coditio for the rows ot beig multiples, eve if oe or both of a, b are zero, ad so by the previous theorem this is the correct coditio for kowig the iverse matrix exists Remark: Determiats are defied for square matrices A ad they determie whether or ot the iverse matrices exist, (ie whether the reduced row echelo form of A is the idetity matrix) Ad whe there are aalogous (more complicated) magic formulas for the iverse matrices, that geeralize the oe above for This is sectio 36 material that we'll discuss carefully o Moday ad Tuesday

3 Math 5-4 Week 6 otes Midterm exam this Friday Feb 7! Mo Feb Matrix iverses; matrix determiats Fiish last Friday's otes: how to fid matrix iverses whe they exist, ad how to figure out whe they do't exist This will lead aturally ito determiats, sectio 36, i this week's otes Determiats are scalars defied for square matrices A ad they always determie whether or ot the iverse matrix A exists, (ie whether the reduced row echelo form of A is the idetity matrix) It turs out that the determiat of A is o-zero if ad oly if A exists The determiat of a matrix a is defied to be the umber a ; determiats of matrices are defied as i Friday's otes; ad i geeral determiats for matrices are defied recursively, i terms of determiats of submatrices: Defiitio: Let A The the determiat of A, writte det A or A, is defied by det A j a j j M j j a j C j Here M j is the determiat of the matrix obtaied from A by deletig the first row ad the j th colum, ad C j is simply j M j More geerally, the determiat of the matrix obtaied by deletig row i ad colum j from A is called the i j Mior M i j of A, ad C i j i j M i j is called the i j Cofactor of A Theorem: (proof is i text appedix) det A ce computed by expadig across ay row, say row i: det A j or by expadig dow ay colum, say colum j: det A i i j M i j i j M i j j i C i j C i j

4 Exercise a) Let A : 3 Compute det A usig the defiitio 5 6 b) Verify that the matrix of all the cofactors of A is give by C i j 3 6 The expad 5 3 det A dow various colums ad rows usig the factors ad C i j cofactors Verify that you always get the same value for det A Notice that i each case you are takig the dot product of a row (or colum) of A with the correspodig row (or colum) of the cofactor matrix c) What happes if you take dot products betwee a row of A ad a differet row of C i j? A colum of A ad a differet colum of C i j? The aswer may seem magic

5 Exercise ) Compute the followig determiats by beig clever about which rows or colums to use: a) ; b) Exercise 3) Explai why it is always true that for a upper triagular matrix (as i a), or for a lower triagular matrix (as i b), the determiat is always just the product of the diagoal etries

6 Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts, which track how elemetary row operatios affect determiats: (a) Swappig ay two rows chages the sig of the determiat proof: This is clear for matrices, sice ad bc, cb ad For 3 3 determiats, expad across the row ot beig swapped, ad use the swap property to deduce the result Prove the geeral result by iductio: oce it's true for matrices you ca prove it for ay matrix, by expadig across a row that was't swapped, ad applyig the result (b) Thus, if two rows i a matrix are the same, the determiat of the matrix must be zero: o the oe had, swappig those two rows leaves the matrix ad its determiat uchaged; o the other had, by (a) the determiat chages its sig The oly way this is possible is if the determiat is zero (a) If you factor a costat out of a row, the you factor the same costat out of the determiat Precisely, usig i ith row of A, ad writig * i i : i c : i * c : i * proof: expad across the i th row, otig that the correspodig cofactors do't chage, sice they're computed by deletig the i th row to get the correspodig miors: det A j C i j j * c a Ci i j j c j * a Ci i j j et A * (b) Combiig (a) with (b), we see that if oe row i A is a scalar multiple of aother, the det A (3) If you replace row i of A by its sum with a multiple of aother row, the the determiat is uchaged! Expad across the i th row: