Linear Algebra Notes by David A. Santos

By David A. Santos

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Solution: ◮ In Z2 every symmetric matrix is also anti-symmetric,   since −x = x. Thus it is enough 0 to take a non-symmetric matrix, for example, take A =   0 1 . 4 Let (A, B) ∈ (M2×2 (R))2 be symmetric matrices. Must their product AB be symmetric? Prove or disprove! 5 Given square matrices (A, B) (M7×7 (R))2 such that tr A2 = tr B2 = 1, and (A − B)2 = 3I7 , as the sum of two 3×3 matrices E1 , E2 , with tr (E2 ) = 10. 2 Give an example of two matrices A ∈ M2×2 (R) and B ∈ M2×2 (R) that simultaneously satisfy the following properties:     0 0 0 0 .

Proof: Expanding the product (In + (λ − 1F )Eii )(In + (λ−1 − 1F )Eii ) = In + (λ − 1F )Eii +(λ−1 − 1F )Eii +(λ − 1F )(λ−1 − 1F )Eii = In + (λ − 1F )Eii +(λ−1 − 1F )Eii +(λ − 1F )(λ−1 − 1F ))Eii = In + (λ − 1F + λ−1 − 1F + 1F −λ − λ−1 − 1F ))Eii = In , proving the assertion. ❑ 39 Matrix Inversion 132 Example By Theorem 131, we have  1   0    0  0 0  1    2 0  0   0 1 0  0   0  1 0      0  = 0 1     1 0 0 1 2 0 0   0 .   1 Repeated applications of Theorem 131 gives the following corollary.

1 0 0 0 0 0 ··· 0 1 −1 1 0 ··· 0 0 0 1 ··· 0 .. .. 0 0 ··· ··· ··· ···  0    0    , 0   ..     1 whence  A−1 1    0    = 0   .  ..    0 −1 1 0 ··· −1 · · · 0 1 .. .. 0 0 ··· ··· ···  0    0    , 0   ..     1 that is, the inverse of A has 1’s on the diagonal and −1’s on the superdiagonal. ◭ 162 Theorem Let A ∈ Mn×n (F) be a triangular matrix such that a11 a22 · · · ann = 0F . Then A is invertible. Proof: Since the entry akk = 0F we multiply the k-th row by a−1 kk and then proceed to subtract the appropriate multiples of the preceding k − 1 rows at each stage.

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