Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rrr} -1 & 4 & 0 \\ 3 & -2 & 2 \\ 5 & 4 & -1 \\ 0 & 8 & -6 \\ -4 & -1 & 0 \end{array}\right], B=\left[\begin{array}{rrr} -3 & 5 & 1 \\ 2 & -4 & -7 \\ 10 & -9 & -1 \\ 3 & 2 & -4 \\ 0 & 1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

\((a) A+B = \begin{bmatrix} -4 & 9 & 1 \ 5 & -6 & -5 \ 15 & -5 & -2 \ 3 & 10 & -10 \ -4 & 0 & -2 \ \end{bmatrix}, (b) A-B = \begin{bmatrix} 2 & -1 & -1 \ 1 & 2 & 9 \ -5 & 13 & 0 \ -3 & 6 & 2 \ -4 & -2 & 2 \ \end{bmatrix}, (c) 3*A = \begin{bmatrix} -3 & 12 & 0 \ 9 & -6 & 6 \ 15 & 12 & -3 \ 0 & 24 & -18 \ -12 & -3 & 0 \ \end{bmatrix}, (d) 3*A - 2*B = \begin{bmatrix} 3 & 2 & -2 \ 7 & 2 & 20 \ -5 & 42 & -1 \ -6 & 20 & -10 \ -12 & -5 & 4 \ \end{bmatrix}\)

Step 1: Perform Addition

Since both matrices are of the same size, it's possible to add matrices A and B. To add matrices, we simply add the corresponding elements in the matrices. Thus:\(A + B = \begin{bmatrix} -1-3 & 4+5 & 0+1 \ 3+2 & -2-4 & 2-7 \5+10 & 4-9 & -1-1 \ 0+3 & 8+2 & -6-4 \ -4+0 & -1+1 & 0-2 \\end{bmatrix} = \begin{bmatrix} -4 & 9 & 1 \ 5 & -6 & -5 \15 & -5 & -2 \ 3 & 10 & -10 \ -4 & 0 & -2\end{bmatrix}\)

Step 2: Perform Subtraction

Next, to subtract matrix B from A, we subtract the corresponding elements of A and B. \(A - B = \begin{bmatrix} -1-(-3) & 4-5 & 0-1 \ 3-2 & -2-(-4) & 2-(-7) \5-10 & 4-(-9) & -1-(-1) \ 0-3 & 8-2 & -6-(-4) \ -4-0 & -1-1 & 0-(-2) \\end{bmatrix} = \begin{bmatrix} 2 & -1 & -1 \ 1 & 2 & 9 \-5 & 13 & 0 \ -3 & 6 & 2 \ -4 & -2 & 2\end{bmatrix}\)

Step 3: Perform Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by the scalar. \(3*A = \begin{bmatrix} 3*(-1) & 3*4 & 3*0 \ 3*3 & 3*(-2) & 3*2 \3*5 & 3*4 & 3*(-1) \ 3*0 & 3*8 & 3*(-6) \ 3*(-4) & 3*(-1) & 3*0 \\end{bmatrix} = \begin{bmatrix} -3 & 12 & 0 \ 9 & -6 & 6 \15 & 12 & -3 \ 0 & 24 & -18 \ -12 & -3 & 0\end{bmatrix}\)

Step 4: Perform Arithmetic Combination

Now apply combination of operations as requested. \(3A - 2B = 3*A - 2*B = \begin{bmatrix} 3*(-1)-2*(-3) & 3*4-2*5 & 3*0-2*1 \ 3*3-2*2 & 3*(-2)-2*(-4) & 3*2-2*(-7) \3*5-2*10 & 3*4-2*(-9) & 3*(-1)-2*(-1) \ 3*0-2*3 & 3*8-2*2 & 3*(-6)-2*(-4) \ 3*(-4)-2*0 & 3*(-1)-2*1 & 3*0-2*(-2) \\end{bmatrix} = \begin{bmatrix} 3 & 2 & -2 \ 7 & 2 & 20 \-5 & 42 & -1 \ -6 & 20 & -10 \ -12 & -5 & 4\end{bmatrix}\)

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation where two matrices of the same dimensions are added together. Each element of the first matrix is added to the corresponding element of the second matrix.
For matrices \(A\) and \(B\) with the same dimensions, their sum, \(A + B\), is computed by adding each element: \\[ (A + B)_{ij} = A_{ij} + B_{ij} \] \Let's take an example with matrices \(A\) and \(B\) having dimensions \(m \times n\) (where \(m\) is the number of rows and \(n\) is the number of columns). Each element is computed as follows:

• \(C_{11} = A_{11} + B_{11}\)
• \(C_{12} = A_{12} + B_{12}\)
• ...

The resulting matrix is called the sum matrix. It contains the sum of each corresponding pair of elements from \(A\) and \(B\). This operation is meant for matrices of the identical size only.
In the exercise, since matrices \(A\) and \(B\) both have the same dimensions, each element of \(A\) was added to the corresponding element in \(B\) to find \(A + B\).

Matrix Subtraction

Matrix subtraction, like addition, involves matrices of identical dimensions where each corresponding element of one matrix is subtracted from the other.
For two matrices \(A\) and \(B\) with the same size, \(A - B\) is computed using:
\[ (A - B)_{ij} = A_{ij} - B_{ij} \] \This means you subtract element \(B_{ij}\) from \(A_{ij}\) for each element in the matrix.

• \(D_{11} = A_{11} - B_{11}\)
• \(D_{12} = A_{12} - B_{12}\)
• ...

Consider this operation as similar to adding negative values. The producing matrix contains the differences of corresponding elements.
For the given matrices \(A\) and \(B\) in the task, subtracting each corresponding element of \(B\) from \(A\) resulted in the matrix \(A - B\).

Scalar Multiplication

Scalar multiplication consists of multiplying each element of a matrix by a scalar value. This changes every element of the matrix proportionally.
To perform scalar multiplication to a matrix \(A\) by a scalar \(k\), each element \(A_{ij}\) is multiplied by \(k\):

• \((kA)_{11} = k \times A_{11}\)
• \((kA)_{12} = k \times A_{12}\)
• ...

This is expressed as:\[ kA = \begin{bmatrix} k \cdot A_{11} & k \cdot A_{12} & \cdots k \cdot A_{21} & k \cdot A_{22} & \cdots \vdots & \vdots & \ddots \end{bmatrix} \] \In the example, multiplying matrix \(A\) by 3 means each element of \(A\) is tripled. This simplified multiplication affects the entire matrix equally, resulting in the scaled matrix \(3A\).

Matrix Arithmetic Combination

Matrix arithmetic combinations involve carrying out more than one operation at a time, like combining addition, subtraction, and scalar multiplication.
In the exercise, the expression \(3A - 2B\) incorporates scalar multiplication and subtraction. Here's a step-by-step breakdown:

• First, multiply matrix \(A\) by scalar 3, resulting in matrix \(3A\).
• Then, multiply matrix \(B\) by scalar 2, resulting in matrix \(2B\).
• Finally, subtract the resulting \(2B\) from \(3A\) to find \(3A - 2B\).

This showcases how these matrix operations can be combined effectively. It's crucial to follow the order of operations properly to achieve the desired result, especially when dealing with scalars and matrices simultaneously.
This complex task demonstrates how matrices can be manipulated via different arithmetic rules, offering flexible and dynamic ways to process data within matrix structures.