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If B and C are non-singular matrices and...

If `B and C` are non-singular matrices and `O` is null matrix, then show that `[[A, B],[ C ,O]]^(-1)=[[O, C^(-1)],[B^(-1),-B^-1A C^(-1)]]dot`

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To show that the inverse of the block matrix \(\begin{bmatrix} A & B \\ C & O \end{bmatrix}\) is given by \(\begin{bmatrix} O & C^{-1} \\ B^{-1} & -B^{-1} A C^{-1} \end{bmatrix}\), we will follow these steps: ### Step 1: Define the Matrix Let \( X = \begin{bmatrix} A & B \\ C & O \end{bmatrix} \). ### Step 2: Calculate the Determinant of \(X\) The determinant of a block matrix of the form \(\begin{bmatrix} P & Q \\ R & S \end{bmatrix}\) can be calculated using the formula: \[ ...
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