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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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