Problem 73 - 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. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$A+c B$$

Step-by-Step Solution

Answer

The result of the operation \(A+cB\) is the matrix \(\left[\begin{array}{rrr}3 & -6 & 0 \ 3 & 3 & 0\end{array}\right]\)

Step 1: Perform Scalar Multiplication

Scalar multiplication is performed by multiplying every entry of the matrix by the scalar. So multiply every entry in matrix B by the scalar c=-2: \[cB = -2\left[\begin{array}{rrr}-1 & 4 & -1 \ -2 & -1 & 0\end{array}\right] = \[\begin{array}{rrr}2 & -8 & 2 \ 4 & 2 & 0\end{array}\right]\]

Step 2: Perform Matrix Addition

Matrix addition is performed element-wise, meaning that each element in the first matrix is added to the corresponding element in the second matrix. Add the resulting matrix from step 1 to matrix A: \[A + cB = \left[\begin{array}{rrr}1 & 2 & -2 \ -1 & 1 & 0\end{array}\right] + \left[\begin{array}{rrr}2 & -8 & 2 \ 4 & 2 & 0\end{array}\right] = \left[\begin{array}{rrr}3 & -6 & 0 \ 3 & 3 & 0\end{array}\right]\]

Step 3: Interpret the result

The resulting matrix is a new 2x3 matrix that was obtained by adding a scalar multiple of matrix B to matrix A. Each element in the resulting matrix corresponds to the sum of the corresponding elements in matrix A and the scalar multiple of matrix B.

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic yet crucial matrix operation. It involves multiplying every element of a matrix by a fixed number, known as the scalar. This process changes the size, but not the shape, of the matrix.

Let's break it down using our example: we need to multiply every element of matrix \[ B = \begin{bmatrix} -1 & 4 & -1 \ -2 & -1 & 0 \end{bmatrix} \]by the scalar \( c = -2 \). Each number inside the matrix \( B \) gets multiplied by \(-2\), like so:

• -1 becomes 2 (since \((-2) \times (-1) = 2\))
• 4 becomes -8 (since \((-2) \times 4 = -8\))
• -1 becomes 2 again (since \((-2) \times (-1) = 2\))
• -2 becomes 4 (since \((-2) \times (-2) = 4\))
• -1 becomes 2
• 0 remains 0

This gives us the new matrix: \[ cB = \begin{bmatrix} 2 & -8 & 2 \ 4 & 2 & 0 \end{bmatrix} \]Scalar multiplication is straightforward, but it's essential since it is often a stepping stone to more complex matrix operations.

Matrix Addition

Matrix addition allows you to combine two matrices. However, this works only if both matrices have the exact same dimensions. They must have the same number of rows and columns. The addition is carried out element-wise, meaning that you add corresponding elements together.

In our step-by-step example, we add the scalar-multiplied matrix \( cB \) to matrix \( A \):\[ A = \begin{bmatrix} 1 & 2 & -2 \ -1 & 1 & 0 \end{bmatrix} \]\[ cB = \begin{bmatrix} 2 & -8 & 2 \ 4 & 2 & 0 \end{bmatrix} \]Each element in these matrices is added like this:

• (1 + 2 = 3)
• (2 + (-8) = -6)
• ((-2) + 2 = 0)
• ((-1) + 4 = 3)
• (1 + 2 = 3)
• (0 + 0 = 0)

So, the resulting matrix is:\[ A + cB = \begin{bmatrix} 3 & -6 & 0 \ 3 & 3 & 0 \end{bmatrix} \]Matrix addition is straightforward when the dimensions match, and it is particularly useful in numerous applications like equilibrium calculations or in computer graphics.

Element-wise Operations

Element-wise operations are a fundamental concept in matrix mathematics and are used extensively in various computational tasks and applications. These operations, including both addition and multiplication, involve independently performing calculations on corresponding elements of matrices.

While in our example, we emphasized element-wise addition when combining matrices, element-wise operations can include multiplication and subtraction, provided the matrices share the same size. Each operation is performed at the index level on every pair of corresponding elements from the involved matrices.

• Addition: As observed, each element in one matrix is added directly to the corresponding element in another, resulting in a new matrix of the same dimensions.
• Multiplication: In cases of element-wise multiplication (also known as the Hadamard product), each element of one matrix is multiplied by the corresponding element of another.

These operations allow for a wide range of matrix transformations and computations, serving as the basis for complex calculations in fields like machine learning, statistics, and physics.