Problem 18 - College Algebra with Modeling and Visualization

Question

If possible, find each of the following. (a) \(A+B\) (b) \(3 A\) (c) \(2 A-3 B\) $$A=\left[\begin{array}{rrr}6 & 2 & 9 \\\3 & -2 & 0 \\\\-1 & 4 & 8\end{array}\right]$$ $$B=\left[\begin{array}{rrr}1 & 0 & -1 \\\3 & 0 & 7 \\\0 & -2 & -5\end{array}\right]$$

Step-by-Step Solution

Answer

(a) \(A+B = \begin{pmatrix} 7 & 2 & 8 \\ 6 & -2 & 7 \\ -1 & 2 & 3 \end{pmatrix}\) (b) \(3A = \begin{pmatrix} 18 & 6 & 27 \\ 9 & -6 & 0 \\ -3 & 12 & 24 \end{pmatrix}\) (c) \(2A-3B = \begin{pmatrix} 9 & 4 & 21 \\ -3 & -4 & -21 \\ -2 & 14 & 31 \end{pmatrix}\)

Step 1: Addition of Matrices

To find \(A+B\), add the corresponding elements of matrices \(A\) and \(B\) together. \[ A+B = \begin{pmatrix} 6+1 & 2+0 & 9+(-1) \ 3+3 & -2+0 & 0+7 \ -1+0 & 4+(-2) & 8+(-5) \end{pmatrix} = \begin{pmatrix} 7 & 2 & 8 \ 6 & -2 & 7 \ -1 & 2 & 3 \end{pmatrix} \]

Step 2: Scalar Multiplication of a Matrix

To find \(3A\), multiply each element in matrix \(A\) by 3. \[ 3A = 3 \times \begin{pmatrix} 6 & 2 & 9 \ 3 & -2 & 0 \ -1 & 4 & 8 \end{pmatrix} = \begin{pmatrix} 18 & 6 & 27 \ 9 & -6 & 0 \ -3 & 12 & 24 \end{pmatrix} \]

Step 3: Linear Combination of Matrices

To find \(2A - 3B\), first compute \(2A\) by multiplying each element of \(A\) by 2, and compute \(3B\) by multiplying each element of \(B\) by 3. Then subtract \(3B\) from \(2A\). \[ 2A = 2 \times \begin{pmatrix} 6 & 2 & 9 \ 3 & -2 & 0 \ -1 & 4 & 8 \end{pmatrix} = \begin{pmatrix} 12 & 4 & 18 \ 6 & -4 & 0 \ -2 & 8 & 16 \end{pmatrix} \] \[ 3B = 3 \times \begin{pmatrix} 1 & 0 & -1 \ 3 & 0 & 7 \ 0 & -2 & -5 \end{pmatrix} = \begin{pmatrix} 3 & 0 & -3 \ 9 & 0 & 21 \ 0 & -6 & -15 \end{pmatrix} \] \[ 2A - 3B = \begin{pmatrix} 12 & 4 & 18 \ 6 & -4 & 0 \ -2 & 8 & 16 \end{pmatrix} - \begin{pmatrix} 3 & 0 & -3 \ 9 & 0 & 21 \ 0 & -6 & -15 \end{pmatrix} = \begin{pmatrix} 9 & 4 & 21 \ -3 & -4 & -21 \ -2 & 14 & 31 \end{pmatrix} \]

Key Concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding elements. This operation is only possible when both matrices have the same dimensions. Imagine matrices as grids of numbers; you can only add them if they are the same size.
To perform the matrix addition, simply:

• Identify each pair of corresponding elements in the matrices.
• Add these elements together to form a new matrix.

For example, consider matrices \(A\) and \(B\), both having 3 rows and 3 columns. To find \(A+B\), add each element in \(A\) with the element at the same location in \(B\).
Mathematically, the sum of matrices \(A\) and \(B\) can be represented as:
\[A + B = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33} \ \end{pmatrix}\]
Once you add all corresponding elements, you end up with a new matrix that reflects the combined values of \(A\) and \(B\).

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant number known as a scalar. This operation is one of the simplest ways to transform a matrix and scales its elements by the given scalar.
To perform scalar multiplication:

• Choose your scalar, say \(k\).
• Multiply every element in the matrix by \(k\).

For example, to calculate \(3A\) where \(A\) is a matrix, simply multiply each of its elements by 3.
Mathematically, the scalar multiplication of matrix \(A\) by 3 is expressed as:
\[3A = \begin{pmatrix} 3 imes a_{11} & 3 imes a_{12} & 3 imes a_{13} \3 imes a_{21} & 3 imes a_{22} & 3 imes a_{23} \3 imes a_{31} & 3 imes a_{32} & 3 imes a_{33} \ \end{pmatrix}\]
This results in a new matrix where each value is scaled by the factor of 3.

Linear Combination of Matrices

A linear combination of matrices involves creating a new matrix by combining two matrices with scalars. This operation is useful in solving linear equations and performing matrix transformations.
To perform a linear combination, you need:

• Two matrices, say \(A\) and \(B\).
• Two scalars, for example \(m\) and \(n\).
• The expression \(mA + nB\).

The process includes scalar multiplication of each matrix and then performing matrix addition or subtraction.
Consider the operation \(2A - 3B\). Begin by multiplying matrix \(A\) by 2 and matrix \(B\) by 3. Then subtract the resulting matrices. The mathematical expression for this operation is:
\[mA + nB = m \times \begin{pmatrix} a_{ij} \end{pmatrix} + n \times \begin{pmatrix} b_{ij} \end{pmatrix}\]
Therefore, the linear combination of matrices results in a new matrix that is a weighted mix of the original matrices, shifting their values according to the specified scalars.