Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$-3\left[\begin{array}{rr}3 & 8 \\ -1 & -9\end{array}\right]+5\left[\begin{array}{rr}4 & -8 \\ 1 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The resultant matrix is \( \begin{bmatrix} 11 & -64 \\ 8 & 57 \end{bmatrix} \).

Step 1: Scalar Multiplication of the First Matrix

Multiply each element in the first matrix by \ -3: \[-3 \times \begin{bmatrix} 3 & 8 \-1 & -9 \end{bmatrix} = \begin{bmatrix} -3 \times 3 & -3 \times 8 \-3 \times (-1) & -3 \times (-9) \end{bmatrix} = \begin{bmatrix} -9 & -24 \3 & 27 \end{bmatrix}\]

Step 2: Scalar Multiplication of the Second Matrix

Multiply each element in the second matrix by 5: \[5 \times \begin{bmatrix} 4 & -8 \1 & 6 \end{bmatrix} = \begin{bmatrix} 5 \times 4 & 5 \times (-8) \5 \times 1 & 5 \times 6 \end{bmatrix} = \begin{bmatrix} 20 & -40 \5 & 30 \end{bmatrix}\]

Step 3: Add the Resultant Matrices

Add the two matrices obtained from the previous steps element-wise:\[\begin{bmatrix} -9 & -24 \3 & 27 \end{bmatrix} + \begin{bmatrix} 20 & -40 \5 & 30 \end{bmatrix} = \begin{bmatrix} -9 + 20 & -24 + (-40) \3 + 5 & 27 + 30 \end{bmatrix}\]Simplify to get:\[\begin{bmatrix} 11 & -64 \8 & 57 \end{bmatrix}\]

Key Concepts

Scalar Multiplication

Scalar multiplication in matrices involves multiplying every element of a matrix by a fixed number, known as a scalar. In this process, each element within the matrix is scaled by the scalar, resulting in a new matrix. Let's dig deeper into this concept using our example.

• Consider a matrix, \( A = \begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix} \), and a scalar, \(-3\). To apply scalar multiplication, multiply \(-3\) with each element of \( A \).
• Here's how it looks: \(-3 \times \begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix} = \begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix}\).

This operation is straightforward. Just multiply the scalar by each matrix element, keeping their position unchanged. This simplicity makes scalar multiplication a fundamental operation in linear algebra.

Matrix Addition

Matrix addition is another basic operation in matrix mathematics. It involves adding corresponding elements of two matrices to create a new matrix. However, it's crucial that the matrices have the same dimensions—same number of rows and columns—for this to be valid.

• In our exercise, we had the results from multiplying by scalars: \( \begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix} \) and \( \begin{bmatrix} 20 & -40 \ 5 & 30 \end{bmatrix} \).
• We simply add them element by element: \( \begin{bmatrix} -9+20 & -24+(-40) \ 3+5 & 27+30 \end{bmatrix} = \begin{bmatrix} 11 & -64 \ 8 & 57 \end{bmatrix} \).

It's that easy! As long as you ensure that both matrices are of equal size, matrix addition simply involves adding the numbers that appear in the same position.

Elementary Algebra

Elementary algebra forms the foundation for understanding operations involving matrices. It provides the basic arithmetic rules and properties that we use in more complex mathematical operations like those involving matrices.

• Scalar multiplication and matrix addition both rely on basic arithmetic—addition, subtraction, and multiplication.
• These operations are called 'elementary' because they apply standard rules you learned when first studying algebra, like the distributive property and associative property, to matrices.

Grasping elementary algebra ensures you can manipulate matrices effectively, gaining the ability to solve more complex mathematical problems. Whether it’s understanding how to distribute a scalar across a matrix or adding matrices element-wise, these basic algebra concepts are crucial.