By Alexander S. Poznyak (Auth.)

ISBN-10: 0080446744

ISBN-13: 9780080446745

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3. 4. If All and A22 are square matrices then 1. for [All O A -- LA21 A22 it follows that det A = (det All) (det A22) 2. 5. (n+l)p (det A12) (det A21) If A is block-diagonal, that is, A - diag [All A22" 9 9Ann ] then 1. 1. 16) Proof Notice that for the case when det A # 0, we have IA D I' CA -1 ~ I Z O OD-CA-1B 1 I 0 A-1B 1 I which leads directly to the first formula. The second formula, when det D # 0, may be proven by the analogous way taking into account the decomposition Matrices and matrix operations IAC DB]-[I0 BD-1I Ez ~ olli 0 D D-1C For proofs of formulas three and four see Gantmacher (1990).

Adj()~In• - A) - Bn_l)~ n-1 + Bn_2~, n-2 + . . - f - BI~. In• -- A ) ( B n - 1 ) f -1 -+- Bn_2~. n-2 -+-... + BI)~ + Bo) = (~n + al) n-1 + . . + an-1)~ + an)In• Comparing coefficients, we obtain Bn-1 - In• B n - 2 - A Bn_ t -- a l In • Bo -- A B I = an-11n• - A B o -- anln• Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 54 Multiplying the first of these equalities by A n, the second one by A n-l, the jth by A n-j+l and adding them together yields 0 -- A n + al An-1 -+-...

F - BI~. In• -- A ) ( B n - 1 ) f -1 -+- Bn_2~. n-2 -+-... + BI)~ + Bo) = (~n + al) n-1 + . . + an-1)~ + an)In• Comparing coefficients, we obtain Bn-1 - In• B n - 2 - A Bn_ t -- a l In • Bo -- A B I = an-11n• - A B o -- anln• Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 54 Multiplying the first of these equalities by A n, the second one by A n-l, the jth by A n-j+l and adding them together yields 0 -- A n + al An-1 -+-... 25). 4. If A -1 exists (A is nonsingular), then A -1 = __~ (A n-1 -+-al An-2 -+-...

### Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques by Alexander S. Poznyak (Auth.)

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