InvDet

# InvDet - Math 17B Kouba Inverses and Determinants of...

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Unformatted text preview: Math 17B Kouba Inverses and Determinants of Matrices DEFINITION : Let A be an n a: n matrix. Matrix A‘1 is the inverse of matrix A if AA‘1 : A‘lA = In , the n x 71 identity matrix. We say that matrix A is invertible . 57 23 (3 3W2 27)=(5 9) (—32 "57)(3 3H?) ‘3), so that A is invertible and A‘1 2 (:32 :57) . 3—7 EXAMPLE 1: Let A = ( -2 5 ) . Consider matrix ( COMM [\Dl—‘M 1 —I 0 1 EXAMPLE 2: Let A = —~1 . Consider matrix 5/3 1/3 —4/3 . Then 1 —1/3 —2/3 2/3 1 —1 0 1 1 0 0 —1 5/3 1/3 —4/3 = 0 1 0 1 —1/3 —2/3 2/3 0 0 1 —1 0 1 2 2 1 1 0 0 (5/3 1/3 4/3) 2 1 —1>=(0 1 0), —1/3 —2/3 2/3 3 2 1 0 0 1 —1 0 1 so that A is invertible and A‘1 = 5/3 1/3 —4/3 . —1/3 —2/3 2/3 CONN [OHM and HOW' TO FIND INVERSES : To ﬁnd A”1 for matrix A 2 Form matrix [A : In] . 1.) 2.) Use matrix reduction rules to create matrix [In : B] . 3.) Then B = A‘1 . NOTE : Not all 'n x n matrices have inverses. EXAMPLE 3: Find the inverse of each matrix. 1 1)A__57_>5710 11'1—2 111—2 “23 23 01"’23 01"“01l—25 10 3 ——7 _1_ 3 —7 ~(01 _2 5), sothat A —(_2 5). DETERMINANTS for 2 a: 2 MATRICES DEFINITION : Let A = (:11 :12 ) . The determinant of matrix A is the number given 21 22 by an (112 det = (111 'a22 —‘ (1124121 021 022 EXAMPLE 4; det :11) = (1)(4)— (—1)(3) =4+3= 7. EXAMPLE 5; det<_41 ‘82) = (—1)(8) — (—2)(4) = —8 + 8 = 0. THEOREM : Matrix A is invertible (nonsingular) if and only if detA : 0 . EXAMPLE 6: Matrix A in EXAMPLE 4 is invertible since detA # 0 . Matrix A in EXAMPLE 5 is NOT invertible since detA : 0 . ...
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InvDet - Math 17B Kouba Inverses and Determinants of...

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