Math Problem Statement
Find formulas for X, Y, and Z in terms of A, B, and C given the matrix equation: [X 0; Y Z] [A 0; B C] = [I 0; 0 I].
Solution
To solve for , , and in terms of , , and , let's analyze the matrix equation provided. The equation is:
\begin{bmatrix} A & 0 \\ B & C \end{bmatrix} = \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix}$$ ### Step 1: Matrix multiplication We multiply the two matrices on the left-hand side: $$\begin{bmatrix} X & 0 \\ Y & Z \end{bmatrix} \begin{bmatrix} A & 0 \\ B & C \end{bmatrix} = \begin{bmatrix} X A & 0 \\ Y A + Z B & Z C \end{bmatrix}$$ ### Step 2: Set equal to the identity matrix We now set this product equal to the right-hand side, which is the identity matrix: $$\begin{bmatrix} X A & 0 \\ Y A + Z B & Z C \end{bmatrix} = \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix}$$ This gives us two systems of equations by comparing the corresponding entries: 1. $$X A = I$$ 2. $$Y A + Z B = 0$$ 3. $$Z C = I$$ ### Step 3: Solve the equations 1. From $$X A = I$$, we solve for $$X$$: $$X = A^{-1}$$ assuming that $$A$$ is invertible. 2. From $$Z C = I$$, we solve for $$Z$$: $$Z = C^{-1}$$ assuming that $$C$$ is invertible. 3. Substitute $$X = A^{-1}$$ and $$Z = C^{-1}$$ into $$Y A + Z B = 0$$: $$Y A + C^{-1} B = 0$$ Solve for $$Y$$: $$Y A = -C^{-1} B$$ $$Y = -C^{-1} B A^{-1}$$ ### Final Answer: - $$X = A^{-1}$$ - $$Y = -C^{-1} B A^{-1}$$ - $$Z = C^{-1}$$ Would you like more details or have any questions? Here are some follow-up questions to consider: 1. What happens if $$A$$ or $$C$$ are not invertible? 2. How would the solution change if $$B = 0$$? 3. Can you find the inverse of a specific matrix $$A$$? 4. How does matrix multiplication order affect the result? 5. What conditions must be met for these matrices to be invertible? **Tip:** Remember that matrix inversion only exists if the matrix is square and non-singular (i.e., it has full rank).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Multiplication
Matrix Inverses
Formulas
X A = I -> X = A^{-1}
Y A + Z B = 0 -> Y = -C^{-1} B A^{-1}
Z C = I -> Z = C^{-1}
Theorems
Invertible Matrix Theorem
Matrix Multiplication Properties
Suitable Grade Level
University Level (Linear Algebra)
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