Skill: Multiply a matrix by a scalar

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Multiplying a matrix by a scalar means multiplying every entry by the same real number, leaving the matrix’s dimensions unchanged; the scalar scales all entries uniformly, reversing their signs when negative and producing the zero matrix when zero. This operation is distinct from matrix multiplication and supports interpreting linear combinations of matrices. Complex scalars and more abstract algebraic settings are not included.

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Detailed Explanation: Multiply a matrix by a scalar

To multiply a matrix by a scalar, multiply each entry of the matrix by that number. The matrix keeps the same dimensions.

For example, find $-2A$ if

A=\begin{bmatrix} 3 & -1\\ 0 & 4 \end{bmatrix}.

Multiply every entry by $-2$ :

-2A = \begin{bmatrix} (-2)(3) & (-2)(-1)\\ (-2)(0) & (-2)(4) \end{bmatrix} = \begin{bmatrix} -6 & 2\\ 0 & -8 \end{bmatrix}.

The negative scalar changes the signs of the nonzero entries, and the result is still a $2\times 2$ matrix.

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