Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$c A+d B$$

Step-by-Step Solution

Answer

The result of the operation \(cA + dB\) is \(\left[\begin{array}{rrr} 1 & -16 & 7 \ 8 & 1 & 0 \end{array}\right]\)

Step 1: Identify the given matrices and scalars

We are given two matrices \(A=\left[\begin{array}{rrr} 1 & 2 & -2 \ -1 & 1 & 0 \end{array}\right]\) and \(B=\left[\begin{array}{rrr} -1 & 4 & -1 \ -2 & -1 & 0 \end{array}\right]\), and two scalars 'c' and 'd', where \(c=-2\) and \(d=-3\). Our task is to find the result of the operation \(cA + dB\) by hand.

Step 2: Perform scalar multiplication

Scalar multiplication of a matrix involves multiplying every element of a matrix by a scalar. So, we multiply all elements of matrix A by 'c' and all elements of matrix B by 'd'. This gives us two new matrices cA and dB. - For cA: \(-2*\left[\begin{array}{rrr} 1 & 2 & -2 \ -1 & 1 & 0 \end{array}\right] = \left[\begin{array}{rrr} -2 & -4 & 4 \ 2 & -2 & 0 \end{array}\right]\) - For dB: \(-3*\left[\begin{array}{rrr} -1 & 4 & -1 \ -2 & -1 & 0 \end{array}\right] = \left[\begin{array}{rrr} 3 & -12 & 3 \ 6 & 3 & 0 \end{array}\right]\)

Step 3: Perform matrix addition

Matrix addition is defined as adding the corresponding elements in the two matrices. Therefore, the sum of matrices cA and dB can be found by adding the corresponding elements in them. Result: \(\left[\begin{array}{rrr} -2 & -4 & 4 \ 2 & -2 & 0 \end{array}\right] + \left[\begin{array}{rrr} 3 & -12 & 3 \ 6 & 3 & 0 \end{array}\right] = \left[\begin{array}{rrr} 1 & -16 & 7 \ 8 & 1 & 0 \end{array}\right]\)

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra. It involves taking a single number, called a scalar, and multiplying it with each element of a matrix. This operation is simple yet powerful, enabling you to scale and transform data stored within matrices.
To perform scalar multiplication, follow these straightforward steps:

• Select your scalar (a given constant value).
• Multiply every element of the matrix by this scalar.

For example, if you have a matrix \[A = \begin{bmatrix} 1 & 2 & -2 \ -1 & 1 & 0 \end{bmatrix}\] and a scalar \( c = -2 \), you multiply each element of \( A \) by \( -2 \) to get:\[-2 \times A = \begin{bmatrix} -2 \times 1 & -2 \times 2 & -2 \times -2 \-2 \times -1 & -2 \times 1 & -2 \times 0 \end{bmatrix} = \begin{bmatrix} -2 & -4 & 4 \2 & -2 & 0 \end{bmatrix}\]Remember, scalar multiplication will change the magnitude but not the shape of the matrix.

Matrix Addition

Matrix addition is one of the basic operations used to combine matrices of the same dimensions. It involves adding up corresponding elements of the matrices.
Here's how matrix addition works:

• Make sure both matrices have the same dimensions. This means each matrix should have the same number of rows and columns.
• Add the elements that occupy the same position in each matrix.

Consider two matrices:\[A = \begin{bmatrix} -2 & -4 & 4 \2 & -2 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -12 & 3 \6 & 3 & 0 \end{bmatrix}\]To find \( A + B \), simply add each element:\[A + B = \begin{bmatrix} -2+3 & -4+(-12) & 4+3 \2+6 & -2+3 & 0+0 \end{bmatrix} = \begin{bmatrix} 1 & -16 & 7 \8 & 1 & 0 \end{bmatrix}\]Matrix addition is straightforward but requires careful attention to ensure all corresponding elements are correctly added together.

Algebraic Operations

Matrix algebraic operations extend basic arithmetic operations such as addition, subtraction, and multiplication into matrices, allowing complex calculations on sets of numbers arranged in rows and columns.
In matrix algebra, the focus is on:

• Scalar Operations: Multiplying a matrix by a scalar is direct and effective for scaling entire datasets.
• Matrix Addition and Subtraction: These operations combine matrices by adding or subtracting corresponding elements. This requires matrices with identical dimensions.
• Matrix Multiplication: Unlike scalar multiplication, matrix multiplication involves a systematic row-by-column multiplication, combining elements across matrices.

Each of these operations plays a distinct role in matrix algebra:

• Scalar operations are useful for adjusting the scale of data and models.
• Addition/subtraction is necessary for combining or comparing datasets.
• Matrix multiplication allows for powerful transformations and analyses, such as rotating or transforming coordinate systems.

Mastering these algebraic operations is essential for solving problems in various fields, from data science to engineering, because they allow you to manipulate entire datasets quickly and systematically.