In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. X - A = B

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1. Identify the given matrices A and B: A = \( \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix} \), B = \( \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix} \).
2. Understand the matrix equation to solve: \( X - A = B \). The goal is to find matrix \( X \).
3. To isolate \( X \), add matrix \( A \) to both sides of the equation: \( X - A + A = B + A \), which simplifies to \( X = B + A \).
4. Perform matrix addition by adding corresponding elements of matrices \( A \) and \( B \). For each element \( x_{ij} \) in \( X \), calculate \( x_{ij} = a_{ij} + b_{ij} \).
5. Write the resulting matrix \( X \) after addition, which will be the solution to the equation.

Key Concepts

Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same dimensions. To subtract matrix A from matrix X, you subtract each element of A from the corresponding element of X. This operation is essential for solving equations like X - A = B.

Matrix Equation Solving

Solving a matrix equation such as X - A = B requires isolating the matrix variable X. This is done by adding matrix A to both sides, resulting in X = A + B. Understanding how to manipulate matrix equations is crucial for finding unknown matrices.

Matrix Dimensions and Compatibility

For matrix operations like addition or subtraction to be valid, the matrices involved must have the same dimensions. Here, matrices A and B are both 3x2, ensuring that operations like X - A = B and X = A + B are defined and can be performed element-wise.