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Prove that (AB)T=BTAT

verify that |−B|=(−1)n|B|

let E=eA=Taylor(e,A), show that:

  1. A−1=e−A
  2. eBeC=eB+C (if B and C commute)
  3. E is orthogonal if A is antisymmetric

Matrix inverse

Find the inverse of A=(223253108).

Sol:

We begin by forming the matrix (A|I3)=(223100253010108001).

Interchanging the first and third rows of the matrix (A|I3), we obtain the matrix (108001253010223100).

Adding (−2) times the first row of the matrix to its second row, we obtain the matrix (10800105−1301−2223100).

Multiplying the second row of the matrix by 15, we obtain the matrix (10800101−135015−25223100).

Adding (−2) times the first row of the matrix to its third row, we obtain the matrix (10800101−135015−2502−1310−2).

Adding (−2) times the second row of the matrix to its third row, we obtain the matrix (10800101−135015−2500−3951−25−65).

Multiplying the third row of the matrix by (−539), we obtain the matrix (10800101−135015−25001−539239213).

Adding (135) times the third row of the matrix (108001010−13130001−539239213).

Adding (−8) times the third row of the matrix to its first row, we obtain the matrix (1004039−1639−313010−13130001−539239213).

Thus, A−1=(4039−1639−313−13130−539239213).