91Ó°ÊÓ

Matrix addition involves adding corresponding elements of the matrices. Given matrices B and C of the same dimensions, their sum B+C is given by the matrix with elements \( (B+C)_{ij} = b_{ij} + c_{ij} \) where i and j range over the number of rows and columns of the matrices.

Given matrices B and C: \[ B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \text{ and } C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} \] add their corresponding elements to get:\[ B + C = \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix} \]

Multiply matrix A by the resulting matrix (B+C): A\(B+C) = \left[ \begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \cdot \left[ \begin{array}{ll} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{array}\right]. \) Perform the multiplication step-by-step: For the first row-first column element: ewline \[ a_{11}(b_{11} + c_{11}) + a_{12}(b_{21} + c_{21}) \] For the first row-second column element: ewline \[ a_{11}(b_{12} + c_{12}) + a_{12}(b_{22} + c_{22}) \] For the second row-first column element: ewline \[ a_{21}(b_{11} + c_{11}) + a_{22}(b_{21} + c_{21}) \] For the second row-second column element: ewline \[ a_{21}(b_{12} + c_{12}) + a_{22}(b_{22} + c_{22}) \]

Multiply matrix A by matrix B: \[ AB = \left[ \begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \cdot \left[ \begin{array}{ll} b_{11} & b_{12} \ b_{21} & b_{22} \end{array} \right] = \left[ \begin{array}{ll} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{array} \right] \] Similarly multiply matrix A by matrix C: \[ AC = \left[ \begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \cdot \left[ \begin{array}{ll} c_{11} & c_{12} \ c_{21} & c_{22} \end{array} \right] = \left[ \begin{array}{ll} a_{11}c_{11} + a_{12}c_{21} & a_{11}c_{12} + a_{12}c_{22} \ a_{21}c_{11} + a_{22}c_{21} & a_{21}c_{12} + a_{22}c_{22} \end{array} \right] \]

Now add the resulting matrices AB and AC: \[ AB + AC = \begin{bmatrix} (a_{11}b_{11} + a_{12}b_{21}) + (a_{11}c_{11} + a_{12}c_{21}) & (a_{11}b_{12} + a_{12}b_{22}) + (a_{11}c_{12} + a_{12}c_{22}) \ (a_{21}b_{11} + a_{22}b_{21}) + (a_{21}c_{11} + a_{22}c_{21}) & (a_{21}b_{12} + a_{22}b_{22}) + (a_{21}c_{12} + a_{22}c_{22}) \end{bmatrix} \]

Simplify each element of the resulting matrix to see if it matches A(B+C):For the first row-first column, we have:\[ a_{11}b_{11} + a_{12}b_{21} + a_{11}c_{11} + a_{12}c_{21} = a_{11}(b_{11} + c_{11}) + a_{12}(b_{21} + c_{21}) \] Similarly, for the rest of the elements. Since each corresponding element in AB + AC matches those in A(B+C), the equality holds: \[ A(B+C) = AB + AC \] Thus, the given statement is true.