Verified Answer :

  • Understand that scalar multiplication involves multiplying every element in the matrix by the given scalar.
  • Perform the multiplication for each entry in the matrix:
  • Row 1: \(5 \times 2 = 10, 5 \times 3 = 15, 5 \times 4 = 20\)
  • Row 2: \(5 \times 8 = 40, 5 \times (-1) = -5, 5 \times (-7) = -35\)
  • Row 3: \(5 \times 11 = 55, 5 \times 5 = 25, 5 \times (-3) = -15\)
  • State the resulting matrix as: \(\boxed{\begin{bmatrix} 10 & 15 & 20 \ 40 & -5 & -35 \ 55 & 25 & -15 \end{bmatrix}}\).

Explanation

  1. Analysis of Matrix Scalar Multiplication

The problem asks for the product of the scalar \(5\) and a given \(3 \times 3\) matrix. A scalar multiplication of a matrix is defined by multiplying every individual entry within the matrix by the given scalar value. In this case, our matrix is
\[M = \begin{bmatrix} 2 & 3 & 4 \ 8 & -1 & -7 \ 11 & 5 & -3 \end{bmatrix}.\]
Our objective is to compute \(5M\). To do this, we multiply each element \(a_{ij}\) in the matrix \(M\) by \(5\). This will result in a new matrix of the same dimensions where each entry \(b_{ij} = 5 \times a_{ij}\).

  1. Computing the Product Matrix

To compute the resulting matrix, we multiply each element of the original matrix by \(5\):

  • Row 1: \(5 \times 2 = 10\), \(5 \times 3 = 15\), \(5 \times 4 = 20\)
  • Row 2: \(5 \times 8 = 40\), \(5 \times (-1) = -5\), \(5 \times (-7) = -35\)
  • Row 3: \(5 \times 11 = 55\), \(5 \times 5 = 25\), \(5 \times (-3) = -15\)

By substituting these values into their respective positions, we obtain the product matrix.

  1. Final Result

Assembling the calculated values into the final \(3 \times 3\) matrix format:
\[5\begin{bmatrix} 2 & 3 & 4 \ 8 & -1 & -7 \ 11 & 5 & -3 \end{bmatrix} = \begin{bmatrix} 10 & 15 & 20 \ 40 & -5 & -35 \ 55 & 25 & -15 \end{bmatrix}.\]
Thus, the product of the scalar \(5\) and the matrix is
\[\boxed{\begin{bmatrix} 10 & 15 & 20 \ 40 & -5 & -35 \ 55 & 25 & -15 \end{bmatrix}}.\]

Examples

Matrix scalar multiplication is widely used in computer graphics and engineering. For example, if you represent a 3D object's vertices as a matrix, scaling the object by a factor of 5 involves multiplying that matrix by the scalar 5. This operation effectively enlarges the object uniformly in all directions while keeping it centered at the origin, allowing designers to easily resize models in software.

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