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Multiplication of Matrices by a Scalar

Definition

The multiplication of a matrix by a scalar is an operation where every element of the matrix is multiplied by the same scalar value. A scalar is simply a constant (a real number), and when multiplying a matrix by a scalar, we apply the multiplication to each entry of the matrix individually.

Mathematically:

If ( A = [a_{ij}] ) is a matrix and ( k ) is a scalar, the product of ( k ) and matrix ( A ) is given by:

[ (k \cdot A) = \left[ k \cdot a_{ij} \right] ]

Where:

  • ( A = [a_{ij}] ) represents the matrix.
  • ( k ) is the scalar.
  • ( k \cdot a_{ij} ) represents the product of the scalar ( k ) with each element of the matrix ( A ).

The resulting matrix will have the same dimensions as matrix ( A ), but each element will be multiplied by the scalar.

Steps for Scalar Multiplication:

  1. Take the scalar and multiply it by each element of the matrix.
  2. Write the resulting values in the corresponding positions.

Example 1:

Let ( A ) be a 2x2 matrix, and let the scalar ( k = 3 ):

[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ]

Now, multiply each element of matrix ( A ) by the scalar ( k = 3 ):

[ 3 \cdot A = 3 \cdot \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 \cdot 1 & 3 \cdot 2 \ 3 \cdot 3 & 3 \cdot 4 \end{bmatrix} ]

[ 3 \cdot A = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} ]

Thus, the result of multiplying matrix ( A ) by 3 is:

[ \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} ]

Example 2:

Let ( B ) be a 3x3 matrix, and let the scalar ( k = -2 ):

[ B = \begin{bmatrix} 4 & -1 & 7 \ 0 & 5 & 3 \ 2 & -6 & 1 \end{bmatrix} ]

Now, multiply each element of matrix ( B ) by the scalar ( k = -2 ):

[ -2 \cdot B = -2 \cdot \begin{bmatrix} 4 & -1 & 7 \ 0 & 5 & 3 \ 2 & -6 & 1 \end{bmatrix} = \begin{bmatrix} -2 \cdot 4 & -2 \cdot (-1) & -2 \cdot 7 \ -2 \cdot 0 & -2 \cdot 5 & -2 \cdot 3 \ -2 \cdot 2 & -2 \cdot (-6) & -2 \cdot 1 \end{bmatrix} ]

[ -2 \cdot B = \begin{bmatrix} -8 & 2 & -14 \ 0 & -10 & -6 \ -4 & 12 & -2 \end{bmatrix} ]

Thus, the result of multiplying matrix ( B ) by -2 is:

[ \begin{bmatrix} -8 & 2 & -14 \ 0 & -10 & -6 \ -4 & 12 & -2 \end{bmatrix} ]

Conclusion

  • Scalar multiplication of matrices involves multiplying each element of the matrix by a scalar value.
  • The resulting matrix has the same dimensions as the original matrix, but its elements are scaled by the scalar.
  • This operation is useful for scaling matrices in various mathematical and real-world applications.
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