91Ó°ÊÓ

Matrix subtraction involves taking two matrices of the same dimensions and subtracting their corresponding elements. Let's use Matrix B and Matrix C, given as follows:Matrix B:$$ B = \begin{bmatrix} 3 & 9 \ 2 & -2 \ end{bmatrix} $$Matrix C:$$ C = \begin{bmatrix} 8 & 3 \ 0 & -3 \ end{bmatrix} $$To subtract matrix C from matrix B, you carry out the subtraction element by element: $$ B - C = \begin{bmatrix} 3 & 9 \ 2 & -2 \ end{bmatrix} - \begin{bmatrix} 8 & 3 \ 0 & -3 \ end{bmatrix} $$This operation translates to:$$ B - C = \begin{bmatrix} 3 - 8 & 9 - 3 \ 2 - 0 & -2 - (-3) \ end{bmatrix} = \begin{bmatrix} -5 & 6 \ 2 & 1 \ end{bmatrix} $$As you can see, matrix subtraction is straightforward as long as you handle each corresponding element correctly.

Scalar multiplication involves multiplying every element of a matrix by a scalar (a single number). Consider matrix C again:Matrix C:$$ C = \begin{bmatrix} 8 & 3 \ 0 & -3 \ end{bmatrix} $$To perform scalar multiplication by -1, multiply each element of C by -1: $$ -C = -1 \times \begin{bmatrix} 8 & 3 \ 0 & -3 \ end{bmatrix} = \begin{bmatrix} -8 & -3 \ 0 & 3 \ end{bmatrix} $$Every element in matrix C is negated, resulting in matrix -C. This operation is foundational, especially in more complex matrix operations such as matrix addition in this context.

Matrix addition requires adding corresponding elements of two matrices of the same dimensions. Suppose we have Matrix B and the negated Matrix C (denoted as -C):Matrix B:$$ B = \begin{bmatrix} 3 & 9 \ 2 & -2 \ end{bmatrix} $$Matrix -C:$$ -C = \begin{bmatrix} -8 & -3 \ 0 & 3 \ end{bmatrix} $$To add these matrices, we perform element-wise addition: $$ B + (-C) = \begin{bmatrix} 3 + (-8) & 9 + (-3) \ 2 + 0 & -2 + 3 \ end{bmatrix} $$This results in: $$ B + (-C) = \begin{bmatrix} -5 & 6 \ 2 & 1 \ end{bmatrix} $$Interestingly, you will find this is identical to the result of B - C. This illustrates that adding the negative of a matrix is equivalent to subtracting the matrix directly, a useful property in matrix operations.